1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2004 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2004 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 By default, the only documentation that will be built is this tutorial
606 in @file{.info} format. To build the GiNaC tutorial and reference manual
607 in HTML, DVI, PostScript, or PDF formats, use one of
616 Generally, the top-level Makefile runs recursively to the
617 subdirectories. It is therefore safe to go into any subdirectory
618 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
619 @var{target} there in case something went wrong.
622 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
623 @c node-name, next, previous, up
624 @section Installing GiNaC
627 To install GiNaC on your system, simply type
633 As described in the section about configuration the files will be
634 installed in the following directories (the directories will be created
635 if they don't already exist):
640 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
641 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
642 So will @file{libginac.so} unless the configure script was
643 given the option @option{--disable-shared}. The proper symlinks
644 will be established as well.
647 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
648 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
651 All documentation (info) will be stuffed into
652 @file{@var{PREFIX}/share/doc/GiNaC/} (or
653 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
657 For the sake of completeness we will list some other useful make
658 targets: @command{make clean} deletes all files generated by
659 @command{make}, i.e. all the object files. In addition @command{make
660 distclean} removes all files generated by the configuration and
661 @command{make maintainer-clean} goes one step further and deletes files
662 that may require special tools to rebuild (like the @command{libtool}
663 for instance). Finally @command{make uninstall} removes the installed
664 library, header files and documentation@footnote{Uninstallation does not
665 work after you have called @command{make distclean} since the
666 @file{Makefile} is itself generated by the configuration from
667 @file{Makefile.in} and hence deleted by @command{make distclean}. There
668 are two obvious ways out of this dilemma. First, you can run the
669 configuration again with the same @var{PREFIX} thus creating a
670 @file{Makefile} with a working @samp{uninstall} target. Second, you can
671 do it by hand since you now know where all the files went during
675 @node Basic Concepts, Expressions, Installing GiNaC, Top
676 @c node-name, next, previous, up
677 @chapter Basic Concepts
679 This chapter will describe the different fundamental objects that can be
680 handled by GiNaC. But before doing so, it is worthwhile introducing you
681 to the more commonly used class of expressions, representing a flexible
682 meta-class for storing all mathematical objects.
685 * Expressions:: The fundamental GiNaC class.
686 * Automatic evaluation:: Evaluation and canonicalization.
687 * Error handling:: How the library reports errors.
688 * The Class Hierarchy:: Overview of GiNaC's classes.
689 * Symbols:: Symbolic objects.
690 * Numbers:: Numerical objects.
691 * Constants:: Pre-defined constants.
692 * Fundamental containers:: Sums, products and powers.
693 * Lists:: Lists of expressions.
694 * Mathematical functions:: Mathematical functions.
695 * Relations:: Equality, Inequality and all that.
696 * Integrals:: Symbolic integrals.
697 * Matrices:: Matrices.
698 * Indexed objects:: Handling indexed quantities.
699 * Non-commutative objects:: Algebras with non-commutative products.
700 * Hash Maps:: A faster alternative to std::map<>.
704 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
705 @c node-name, next, previous, up
707 @cindex expression (class @code{ex})
710 The most common class of objects a user deals with is the expression
711 @code{ex}, representing a mathematical object like a variable, number,
712 function, sum, product, etc@dots{} Expressions may be put together to form
713 new expressions, passed as arguments to functions, and so on. Here is a
714 little collection of valid expressions:
717 ex MyEx1 = 5; // simple number
718 ex MyEx2 = x + 2*y; // polynomial in x and y
719 ex MyEx3 = (x + 1)/(x - 1); // rational expression
720 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
721 ex MyEx5 = MyEx4 + 1; // similar to above
724 Expressions are handles to other more fundamental objects, that often
725 contain other expressions thus creating a tree of expressions
726 (@xref{Internal Structures}, for particular examples). Most methods on
727 @code{ex} therefore run top-down through such an expression tree. For
728 example, the method @code{has()} scans recursively for occurrences of
729 something inside an expression. Thus, if you have declared @code{MyEx4}
730 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
731 the argument of @code{sin} and hence return @code{true}.
733 The next sections will outline the general picture of GiNaC's class
734 hierarchy and describe the classes of objects that are handled by
737 @subsection Note: Expressions and STL containers
739 GiNaC expressions (@code{ex} objects) have value semantics (they can be
740 assigned, reassigned and copied like integral types) but the operator
741 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
742 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
744 This implies that in order to use expressions in sorted containers such as
745 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
746 comparison predicate. GiNaC provides such a predicate, called
747 @code{ex_is_less}. For example, a set of expressions should be defined
748 as @code{std::set<ex, ex_is_less>}.
750 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
751 don't pose a problem. A @code{std::vector<ex>} works as expected.
753 @xref{Information About Expressions}, for more about comparing and ordering
757 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
758 @c node-name, next, previous, up
759 @section Automatic evaluation and canonicalization of expressions
762 GiNaC performs some automatic transformations on expressions, to simplify
763 them and put them into a canonical form. Some examples:
766 ex MyEx1 = 2*x - 1 + x; // 3*x-1
767 ex MyEx2 = x - x; // 0
768 ex MyEx3 = cos(2*Pi); // 1
769 ex MyEx4 = x*y/x; // y
772 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
773 evaluation}. GiNaC only performs transformations that are
777 at most of complexity
785 algebraically correct, possibly except for a set of measure zero (e.g.
786 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
789 There are two types of automatic transformations in GiNaC that may not
790 behave in an entirely obvious way at first glance:
794 The terms of sums and products (and some other things like the arguments of
795 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
796 into a canonical form that is deterministic, but not lexicographical or in
797 any other way easy to guess (it almost always depends on the number and
798 order of the symbols you define). However, constructing the same expression
799 twice, either implicitly or explicitly, will always result in the same
802 Expressions of the form 'number times sum' are automatically expanded (this
803 has to do with GiNaC's internal representation of sums and products). For
806 ex MyEx5 = 2*(x + y); // 2*x+2*y
807 ex MyEx6 = z*(x + y); // z*(x+y)
811 The general rule is that when you construct expressions, GiNaC automatically
812 creates them in canonical form, which might differ from the form you typed in
813 your program. This may create some awkward looking output (@samp{-y+x} instead
814 of @samp{x-y}) but allows for more efficient operation and usually yields
815 some immediate simplifications.
817 @cindex @code{eval()}
818 Internally, the anonymous evaluator in GiNaC is implemented by the methods
821 ex ex::eval(int level = 0) const;
822 ex basic::eval(int level = 0) const;
825 but unless you are extending GiNaC with your own classes or functions, there
826 should never be any reason to call them explicitly. All GiNaC methods that
827 transform expressions, like @code{subs()} or @code{normal()}, automatically
828 re-evaluate their results.
831 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
832 @c node-name, next, previous, up
833 @section Error handling
835 @cindex @code{pole_error} (class)
837 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
838 generated by GiNaC are subclassed from the standard @code{exception} class
839 defined in the @file{<stdexcept>} header. In addition to the predefined
840 @code{logic_error}, @code{domain_error}, @code{out_of_range},
841 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
842 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
843 exception that gets thrown when trying to evaluate a mathematical function
846 The @code{pole_error} class has a member function
849 int pole_error::degree() const;
852 that returns the order of the singularity (or 0 when the pole is
853 logarithmic or the order is undefined).
855 When using GiNaC it is useful to arrange for exceptions to be caught in
856 the main program even if you don't want to do any special error handling.
857 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
858 default exception handler of your C++ compiler's run-time system which
859 usually only aborts the program without giving any information what went
862 Here is an example for a @code{main()} function that catches and prints
863 exceptions generated by GiNaC:
868 #include <ginac/ginac.h>
870 using namespace GiNaC;
878 @} catch (exception &p) @{
879 cerr << p.what() << endl;
887 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
888 @c node-name, next, previous, up
889 @section The Class Hierarchy
891 GiNaC's class hierarchy consists of several classes representing
892 mathematical objects, all of which (except for @code{ex} and some
893 helpers) are internally derived from one abstract base class called
894 @code{basic}. You do not have to deal with objects of class
895 @code{basic}, instead you'll be dealing with symbols, numbers,
896 containers of expressions and so on.
900 To get an idea about what kinds of symbolic composites may be built we
901 have a look at the most important classes in the class hierarchy and
902 some of the relations among the classes:
904 @image{classhierarchy}
906 The abstract classes shown here (the ones without drop-shadow) are of no
907 interest for the user. They are used internally in order to avoid code
908 duplication if two or more classes derived from them share certain
909 features. An example is @code{expairseq}, a container for a sequence of
910 pairs each consisting of one expression and a number (@code{numeric}).
911 What @emph{is} visible to the user are the derived classes @code{add}
912 and @code{mul}, representing sums and products. @xref{Internal
913 Structures}, where these two classes are described in more detail. The
914 following table shortly summarizes what kinds of mathematical objects
915 are stored in the different classes:
918 @multitable @columnfractions .22 .78
919 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
920 @item @code{constant} @tab Constants like
927 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
928 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
929 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
930 @item @code{ncmul} @tab Products of non-commutative objects
931 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
936 @code{sqrt(}@math{2}@code{)}
939 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
940 @item @code{function} @tab A symbolic function like
947 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
948 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
949 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
950 @item @code{indexed} @tab Indexed object like @math{A_ij}
951 @item @code{tensor} @tab Special tensor like the delta and metric tensors
952 @item @code{idx} @tab Index of an indexed object
953 @item @code{varidx} @tab Index with variance
954 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
955 @item @code{wildcard} @tab Wildcard for pattern matching
956 @item @code{structure} @tab Template for user-defined classes
961 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
962 @c node-name, next, previous, up
964 @cindex @code{symbol} (class)
965 @cindex hierarchy of classes
968 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
969 manipulation what atoms are for chemistry.
971 A typical symbol definition looks like this:
976 This definition actually contains three very different things:
978 @item a C++ variable named @code{x}
979 @item a @code{symbol} object stored in this C++ variable; this object
980 represents the symbol in a GiNaC expression
981 @item the string @code{"x"} which is the name of the symbol, used (almost)
982 exclusively for printing expressions holding the symbol
985 Symbols have an explicit name, supplied as a string during construction,
986 because in C++, variable names can't be used as values, and the C++ compiler
987 throws them away during compilation.
989 It is possible to omit the symbol name in the definition:
994 In this case, GiNaC will assign the symbol an internal, unique name of the
995 form @code{symbolNNN}. This won't affect the usability of the symbol but
996 the output of your calculations will become more readable if you give your
997 symbols sensible names (for intermediate expressions that are only used
998 internally such anonymous symbols can be quite useful, however).
1000 Now, here is one important property of GiNaC that differentiates it from
1001 other computer algebra programs you may have used: GiNaC does @emph{not} use
1002 the names of symbols to tell them apart, but a (hidden) serial number that
1003 is unique for each newly created @code{symbol} object. In you want to use
1004 one and the same symbol in different places in your program, you must only
1005 create one @code{symbol} object and pass that around. If you create another
1006 symbol, even if it has the same name, GiNaC will treat it as a different
1023 // prints "x^6" which looks right, but...
1025 cout << e.degree(x) << endl;
1026 // ...this doesn't work. The symbol "x" here is different from the one
1027 // in f() and in the expression returned by f(). Consequently, it
1032 One possibility to ensure that @code{f()} and @code{main()} use the same
1033 symbol is to pass the symbol as an argument to @code{f()}:
1035 ex f(int n, const ex & x)
1044 // Now, f() uses the same symbol.
1047 cout << e.degree(x) << endl;
1048 // prints "6", as expected
1052 Another possibility would be to define a global symbol @code{x} that is used
1053 by both @code{f()} and @code{main()}. If you are using global symbols and
1054 multiple compilation units you must take special care, however. Suppose
1055 that you have a header file @file{globals.h} in your program that defines
1056 a @code{symbol x("x");}. In this case, every unit that includes
1057 @file{globals.h} would also get its own definition of @code{x} (because
1058 header files are just inlined into the source code by the C++ preprocessor),
1059 and hence you would again end up with multiple equally-named, but different,
1060 symbols. Instead, the @file{globals.h} header should only contain a
1061 @emph{declaration} like @code{extern symbol x;}, with the definition of
1062 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1064 A different approach to ensuring that symbols used in different parts of
1065 your program are identical is to create them with a @emph{factory} function
1068 const symbol & get_symbol(const string & s)
1070 static map<string, symbol> directory;
1071 map<string, symbol>::iterator i = directory.find(s);
1072 if (i != directory.end())
1075 return directory.insert(make_pair(s, symbol(s))).first->second;
1079 This function returns one newly constructed symbol for each name that is
1080 passed in, and it returns the same symbol when called multiple times with
1081 the same name. Using this symbol factory, we can rewrite our example like
1086 return pow(get_symbol("x"), n);
1093 // Both calls of get_symbol("x") yield the same symbol.
1094 cout << e.degree(get_symbol("x")) << endl;
1099 Instead of creating symbols from strings we could also have
1100 @code{get_symbol()} take, for example, an integer number as its argument.
1101 In this case, we would probably want to give the generated symbols names
1102 that include this number, which can be accomplished with the help of an
1103 @code{ostringstream}.
1105 In general, if you're getting weird results from GiNaC such as an expression
1106 @samp{x-x} that is not simplified to zero, you should check your symbol
1109 As we said, the names of symbols primarily serve for purposes of expression
1110 output. But there are actually two instances where GiNaC uses the names for
1111 identifying symbols: When constructing an expression from a string, and when
1112 recreating an expression from an archive (@pxref{Input/Output}).
1114 In addition to its name, a symbol may contain a special string that is used
1117 symbol x("x", "\\Box");
1120 This creates a symbol that is printed as "@code{x}" in normal output, but
1121 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1122 information about the different output formats of expressions in GiNaC).
1123 GiNaC automatically creates proper LaTeX code for symbols having names of
1124 greek letters (@samp{alpha}, @samp{mu}, etc.).
1126 @cindex @code{subs()}
1127 Symbols in GiNaC can't be assigned values. If you need to store results of
1128 calculations and give them a name, use C++ variables of type @code{ex}.
1129 If you want to replace a symbol in an expression with something else, you
1130 can invoke the expression's @code{.subs()} method
1131 (@pxref{Substituting Expressions}).
1133 @cindex @code{realsymbol()}
1134 By default, symbols are expected to stand in for complex values, i.e. they live
1135 in the complex domain. As a consequence, operations like complex conjugation,
1136 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1137 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1138 because of the unknown imaginary part of @code{x}.
1139 On the other hand, if you are sure that your symbols will hold only real values, you
1140 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1141 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1142 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1145 @node Numbers, Constants, Symbols, Basic Concepts
1146 @c node-name, next, previous, up
1148 @cindex @code{numeric} (class)
1154 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1155 The classes therein serve as foundation classes for GiNaC. CLN stands
1156 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1157 In order to find out more about CLN's internals, the reader is referred to
1158 the documentation of that library. @inforef{Introduction, , cln}, for
1159 more information. Suffice to say that it is by itself build on top of
1160 another library, the GNU Multiple Precision library GMP, which is an
1161 extremely fast library for arbitrary long integers and rationals as well
1162 as arbitrary precision floating point numbers. It is very commonly used
1163 by several popular cryptographic applications. CLN extends GMP by
1164 several useful things: First, it introduces the complex number field
1165 over either reals (i.e. floating point numbers with arbitrary precision)
1166 or rationals. Second, it automatically converts rationals to integers
1167 if the denominator is unity and complex numbers to real numbers if the
1168 imaginary part vanishes and also correctly treats algebraic functions.
1169 Third it provides good implementations of state-of-the-art algorithms
1170 for all trigonometric and hyperbolic functions as well as for
1171 calculation of some useful constants.
1173 The user can construct an object of class @code{numeric} in several
1174 ways. The following example shows the four most important constructors.
1175 It uses construction from C-integer, construction of fractions from two
1176 integers, construction from C-float and construction from a string:
1180 #include <ginac/ginac.h>
1181 using namespace GiNaC;
1185 numeric two = 2; // exact integer 2
1186 numeric r(2,3); // exact fraction 2/3
1187 numeric e(2.71828); // floating point number
1188 numeric p = "3.14159265358979323846"; // constructor from string
1189 // Trott's constant in scientific notation:
1190 numeric trott("1.0841015122311136151E-2");
1192 std::cout << two*p << std::endl; // floating point 6.283...
1197 @cindex complex numbers
1198 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1203 numeric z1 = 2-3*I; // exact complex number 2-3i
1204 numeric z2 = 5.9+1.6*I; // complex floating point number
1208 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1209 This would, however, call C's built-in operator @code{/} for integers
1210 first and result in a numeric holding a plain integer 1. @strong{Never
1211 use the operator @code{/} on integers} unless you know exactly what you
1212 are doing! Use the constructor from two integers instead, as shown in
1213 the example above. Writing @code{numeric(1)/2} may look funny but works
1216 @cindex @code{Digits}
1218 We have seen now the distinction between exact numbers and floating
1219 point numbers. Clearly, the user should never have to worry about
1220 dynamically created exact numbers, since their `exactness' always
1221 determines how they ought to be handled, i.e. how `long' they are. The
1222 situation is different for floating point numbers. Their accuracy is
1223 controlled by one @emph{global} variable, called @code{Digits}. (For
1224 those readers who know about Maple: it behaves very much like Maple's
1225 @code{Digits}). All objects of class numeric that are constructed from
1226 then on will be stored with a precision matching that number of decimal
1231 #include <ginac/ginac.h>
1232 using namespace std;
1233 using namespace GiNaC;
1237 numeric three(3.0), one(1.0);
1238 numeric x = one/three;
1240 cout << "in " << Digits << " digits:" << endl;
1242 cout << Pi.evalf() << endl;
1254 The above example prints the following output to screen:
1258 0.33333333333333333334
1259 3.1415926535897932385
1261 0.33333333333333333333333333333333333333333333333333333333333333333334
1262 3.1415926535897932384626433832795028841971693993751058209749445923078
1266 Note that the last number is not necessarily rounded as you would
1267 naively expect it to be rounded in the decimal system. But note also,
1268 that in both cases you got a couple of extra digits. This is because
1269 numbers are internally stored by CLN as chunks of binary digits in order
1270 to match your machine's word size and to not waste precision. Thus, on
1271 architectures with different word size, the above output might even
1272 differ with regard to actually computed digits.
1274 It should be clear that objects of class @code{numeric} should be used
1275 for constructing numbers or for doing arithmetic with them. The objects
1276 one deals with most of the time are the polymorphic expressions @code{ex}.
1278 @subsection Tests on numbers
1280 Once you have declared some numbers, assigned them to expressions and
1281 done some arithmetic with them it is frequently desired to retrieve some
1282 kind of information from them like asking whether that number is
1283 integer, rational, real or complex. For those cases GiNaC provides
1284 several useful methods. (Internally, they fall back to invocations of
1285 certain CLN functions.)
1287 As an example, let's construct some rational number, multiply it with
1288 some multiple of its denominator and test what comes out:
1292 #include <ginac/ginac.h>
1293 using namespace std;
1294 using namespace GiNaC;
1296 // some very important constants:
1297 const numeric twentyone(21);
1298 const numeric ten(10);
1299 const numeric five(5);
1303 numeric answer = twentyone;
1306 cout << answer.is_integer() << endl; // false, it's 21/5
1308 cout << answer.is_integer() << endl; // true, it's 42 now!
1312 Note that the variable @code{answer} is constructed here as an integer
1313 by @code{numeric}'s copy constructor but in an intermediate step it
1314 holds a rational number represented as integer numerator and integer
1315 denominator. When multiplied by 10, the denominator becomes unity and
1316 the result is automatically converted to a pure integer again.
1317 Internally, the underlying CLN is responsible for this behavior and we
1318 refer the reader to CLN's documentation. Suffice to say that
1319 the same behavior applies to complex numbers as well as return values of
1320 certain functions. Complex numbers are automatically converted to real
1321 numbers if the imaginary part becomes zero. The full set of tests that
1322 can be applied is listed in the following table.
1325 @multitable @columnfractions .30 .70
1326 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1327 @item @code{.is_zero()}
1328 @tab @dots{}equal to zero
1329 @item @code{.is_positive()}
1330 @tab @dots{}not complex and greater than 0
1331 @item @code{.is_integer()}
1332 @tab @dots{}a (non-complex) integer
1333 @item @code{.is_pos_integer()}
1334 @tab @dots{}an integer and greater than 0
1335 @item @code{.is_nonneg_integer()}
1336 @tab @dots{}an integer and greater equal 0
1337 @item @code{.is_even()}
1338 @tab @dots{}an even integer
1339 @item @code{.is_odd()}
1340 @tab @dots{}an odd integer
1341 @item @code{.is_prime()}
1342 @tab @dots{}a prime integer (probabilistic primality test)
1343 @item @code{.is_rational()}
1344 @tab @dots{}an exact rational number (integers are rational, too)
1345 @item @code{.is_real()}
1346 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1347 @item @code{.is_cinteger()}
1348 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1349 @item @code{.is_crational()}
1350 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1354 @subsection Numeric functions
1356 The following functions can be applied to @code{numeric} objects and will be
1357 evaluated immediately:
1360 @multitable @columnfractions .30 .70
1361 @item @strong{Name} @tab @strong{Function}
1362 @item @code{inverse(z)}
1363 @tab returns @math{1/z}
1364 @cindex @code{inverse()} (numeric)
1365 @item @code{pow(a, b)}
1366 @tab exponentiation @math{a^b}
1369 @item @code{real(z)}
1371 @cindex @code{real()}
1372 @item @code{imag(z)}
1374 @cindex @code{imag()}
1375 @item @code{csgn(z)}
1376 @tab complex sign (returns an @code{int})
1377 @item @code{numer(z)}
1378 @tab numerator of rational or complex rational number
1379 @item @code{denom(z)}
1380 @tab denominator of rational or complex rational number
1381 @item @code{sqrt(z)}
1383 @item @code{isqrt(n)}
1384 @tab integer square root
1385 @cindex @code{isqrt()}
1392 @item @code{asin(z)}
1394 @item @code{acos(z)}
1396 @item @code{atan(z)}
1397 @tab inverse tangent
1398 @item @code{atan(y, x)}
1399 @tab inverse tangent with two arguments
1400 @item @code{sinh(z)}
1401 @tab hyperbolic sine
1402 @item @code{cosh(z)}
1403 @tab hyperbolic cosine
1404 @item @code{tanh(z)}
1405 @tab hyperbolic tangent
1406 @item @code{asinh(z)}
1407 @tab inverse hyperbolic sine
1408 @item @code{acosh(z)}
1409 @tab inverse hyperbolic cosine
1410 @item @code{atanh(z)}
1411 @tab inverse hyperbolic tangent
1413 @tab exponential function
1415 @tab natural logarithm
1418 @item @code{zeta(z)}
1419 @tab Riemann's zeta function
1420 @item @code{tgamma(z)}
1422 @item @code{lgamma(z)}
1423 @tab logarithm of gamma function
1425 @tab psi (digamma) function
1426 @item @code{psi(n, z)}
1427 @tab derivatives of psi function (polygamma functions)
1428 @item @code{factorial(n)}
1429 @tab factorial function @math{n!}
1430 @item @code{doublefactorial(n)}
1431 @tab double factorial function @math{n!!}
1432 @cindex @code{doublefactorial()}
1433 @item @code{binomial(n, k)}
1434 @tab binomial coefficients
1435 @item @code{bernoulli(n)}
1436 @tab Bernoulli numbers
1437 @cindex @code{bernoulli()}
1438 @item @code{fibonacci(n)}
1439 @tab Fibonacci numbers
1440 @cindex @code{fibonacci()}
1441 @item @code{mod(a, b)}
1442 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1443 @cindex @code{mod()}
1444 @item @code{smod(a, b)}
1445 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1446 @cindex @code{smod()}
1447 @item @code{irem(a, b)}
1448 @tab integer remainder (has the sign of @math{a}, or is zero)
1449 @cindex @code{irem()}
1450 @item @code{irem(a, b, q)}
1451 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1452 @item @code{iquo(a, b)}
1453 @tab integer quotient
1454 @cindex @code{iquo()}
1455 @item @code{iquo(a, b, r)}
1456 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1457 @item @code{gcd(a, b)}
1458 @tab greatest common divisor
1459 @item @code{lcm(a, b)}
1460 @tab least common multiple
1464 Most of these functions are also available as symbolic functions that can be
1465 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1466 as polynomial algorithms.
1468 @subsection Converting numbers
1470 Sometimes it is desirable to convert a @code{numeric} object back to a
1471 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1472 class provides a couple of methods for this purpose:
1474 @cindex @code{to_int()}
1475 @cindex @code{to_long()}
1476 @cindex @code{to_double()}
1477 @cindex @code{to_cl_N()}
1479 int numeric::to_int() const;
1480 long numeric::to_long() const;
1481 double numeric::to_double() const;
1482 cln::cl_N numeric::to_cl_N() const;
1485 @code{to_int()} and @code{to_long()} only work when the number they are
1486 applied on is an exact integer. Otherwise the program will halt with a
1487 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1488 rational number will return a floating-point approximation. Both
1489 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1490 part of complex numbers.
1493 @node Constants, Fundamental containers, Numbers, Basic Concepts
1494 @c node-name, next, previous, up
1496 @cindex @code{constant} (class)
1499 @cindex @code{Catalan}
1500 @cindex @code{Euler}
1501 @cindex @code{evalf()}
1502 Constants behave pretty much like symbols except that they return some
1503 specific number when the method @code{.evalf()} is called.
1505 The predefined known constants are:
1508 @multitable @columnfractions .14 .30 .56
1509 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1511 @tab Archimedes' constant
1512 @tab 3.14159265358979323846264338327950288
1513 @item @code{Catalan}
1514 @tab Catalan's constant
1515 @tab 0.91596559417721901505460351493238411
1517 @tab Euler's (or Euler-Mascheroni) constant
1518 @tab 0.57721566490153286060651209008240243
1523 @node Fundamental containers, Lists, Constants, Basic Concepts
1524 @c node-name, next, previous, up
1525 @section Sums, products and powers
1529 @cindex @code{power}
1531 Simple rational expressions are written down in GiNaC pretty much like
1532 in other CAS or like expressions involving numerical variables in C.
1533 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1534 been overloaded to achieve this goal. When you run the following
1535 code snippet, the constructor for an object of type @code{mul} is
1536 automatically called to hold the product of @code{a} and @code{b} and
1537 then the constructor for an object of type @code{add} is called to hold
1538 the sum of that @code{mul} object and the number one:
1542 symbol a("a"), b("b");
1547 @cindex @code{pow()}
1548 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1549 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1550 construction is necessary since we cannot safely overload the constructor
1551 @code{^} in C++ to construct a @code{power} object. If we did, it would
1552 have several counterintuitive and undesired effects:
1556 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1558 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1559 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1560 interpret this as @code{x^(a^b)}.
1562 Also, expressions involving integer exponents are very frequently used,
1563 which makes it even more dangerous to overload @code{^} since it is then
1564 hard to distinguish between the semantics as exponentiation and the one
1565 for exclusive or. (It would be embarrassing to return @code{1} where one
1566 has requested @code{2^3}.)
1569 @cindex @command{ginsh}
1570 All effects are contrary to mathematical notation and differ from the
1571 way most other CAS handle exponentiation, therefore overloading @code{^}
1572 is ruled out for GiNaC's C++ part. The situation is different in
1573 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1574 that the other frequently used exponentiation operator @code{**} does
1575 not exist at all in C++).
1577 To be somewhat more precise, objects of the three classes described
1578 here, are all containers for other expressions. An object of class
1579 @code{power} is best viewed as a container with two slots, one for the
1580 basis, one for the exponent. All valid GiNaC expressions can be
1581 inserted. However, basic transformations like simplifying
1582 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1583 when this is mathematically possible. If we replace the outer exponent
1584 three in the example by some symbols @code{a}, the simplification is not
1585 safe and will not be performed, since @code{a} might be @code{1/2} and
1588 Objects of type @code{add} and @code{mul} are containers with an
1589 arbitrary number of slots for expressions to be inserted. Again, simple
1590 and safe simplifications are carried out like transforming
1591 @code{3*x+4-x} to @code{2*x+4}.
1594 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1595 @c node-name, next, previous, up
1596 @section Lists of expressions
1597 @cindex @code{lst} (class)
1599 @cindex @code{nops()}
1601 @cindex @code{append()}
1602 @cindex @code{prepend()}
1603 @cindex @code{remove_first()}
1604 @cindex @code{remove_last()}
1605 @cindex @code{remove_all()}
1607 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1608 expressions. They are not as ubiquitous as in many other computer algebra
1609 packages, but are sometimes used to supply a variable number of arguments of
1610 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1611 constructors, so you should have a basic understanding of them.
1613 Lists can be constructed by assigning a comma-separated sequence of
1618 symbol x("x"), y("y");
1621 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1626 There are also constructors that allow direct creation of lists of up to
1627 16 expressions, which is often more convenient but slightly less efficient:
1631 // This produces the same list 'l' as above:
1632 // lst l(x, 2, y, x+y);
1633 // lst l = lst(x, 2, y, x+y);
1637 Use the @code{nops()} method to determine the size (number of expressions) of
1638 a list and the @code{op()} method or the @code{[]} operator to access
1639 individual elements:
1643 cout << l.nops() << endl; // prints '4'
1644 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1648 As with the standard @code{list<T>} container, accessing random elements of a
1649 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1650 sequential access to the elements of a list is possible with the
1651 iterator types provided by the @code{lst} class:
1654 typedef ... lst::const_iterator;
1655 typedef ... lst::const_reverse_iterator;
1656 lst::const_iterator lst::begin() const;
1657 lst::const_iterator lst::end() const;
1658 lst::const_reverse_iterator lst::rbegin() const;
1659 lst::const_reverse_iterator lst::rend() const;
1662 For example, to print the elements of a list individually you can use:
1667 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1672 which is one order faster than
1677 for (size_t i = 0; i < l.nops(); ++i)
1678 cout << l.op(i) << endl;
1682 These iterators also allow you to use some of the algorithms provided by
1683 the C++ standard library:
1687 // print the elements of the list (requires #include <iterator>)
1688 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1690 // sum up the elements of the list (requires #include <numeric>)
1691 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1692 cout << sum << endl; // prints '2+2*x+2*y'
1696 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1697 (the only other one is @code{matrix}). You can modify single elements:
1701 l[1] = 42; // l is now @{x, 42, y, x+y@}
1702 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1706 You can append or prepend an expression to a list with the @code{append()}
1707 and @code{prepend()} methods:
1711 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1712 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1716 You can remove the first or last element of a list with @code{remove_first()}
1717 and @code{remove_last()}:
1721 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1722 l.remove_last(); // l is now @{x, 7, y, x+y@}
1726 You can remove all the elements of a list with @code{remove_all()}:
1730 l.remove_all(); // l is now empty
1734 You can bring the elements of a list into a canonical order with @code{sort()}:
1743 // l1 and l2 are now equal
1747 Finally, you can remove all but the first element of consecutive groups of
1748 elements with @code{unique()}:
1753 l3 = x, 2, 2, 2, y, x+y, y+x;
1754 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1759 @node Mathematical functions, Relations, Lists, Basic Concepts
1760 @c node-name, next, previous, up
1761 @section Mathematical functions
1762 @cindex @code{function} (class)
1763 @cindex trigonometric function
1764 @cindex hyperbolic function
1766 There are quite a number of useful functions hard-wired into GiNaC. For
1767 instance, all trigonometric and hyperbolic functions are implemented
1768 (@xref{Built-in Functions}, for a complete list).
1770 These functions (better called @emph{pseudofunctions}) are all objects
1771 of class @code{function}. They accept one or more expressions as
1772 arguments and return one expression. If the arguments are not
1773 numerical, the evaluation of the function may be halted, as it does in
1774 the next example, showing how a function returns itself twice and
1775 finally an expression that may be really useful:
1777 @cindex Gamma function
1778 @cindex @code{subs()}
1781 symbol x("x"), y("y");
1783 cout << tgamma(foo) << endl;
1784 // -> tgamma(x+(1/2)*y)
1785 ex bar = foo.subs(y==1);
1786 cout << tgamma(bar) << endl;
1788 ex foobar = bar.subs(x==7);
1789 cout << tgamma(foobar) << endl;
1790 // -> (135135/128)*Pi^(1/2)
1794 Besides evaluation most of these functions allow differentiation, series
1795 expansion and so on. Read the next chapter in order to learn more about
1798 It must be noted that these pseudofunctions are created by inline
1799 functions, where the argument list is templated. This means that
1800 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1801 @code{sin(ex(1))} and will therefore not result in a floating point
1802 number. Unless of course the function prototype is explicitly
1803 overridden -- which is the case for arguments of type @code{numeric}
1804 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1805 point number of class @code{numeric} you should call
1806 @code{sin(numeric(1))}. This is almost the same as calling
1807 @code{sin(1).evalf()} except that the latter will return a numeric
1808 wrapped inside an @code{ex}.
1811 @node Relations, Integrals, Mathematical functions, Basic Concepts
1812 @c node-name, next, previous, up
1814 @cindex @code{relational} (class)
1816 Sometimes, a relation holding between two expressions must be stored
1817 somehow. The class @code{relational} is a convenient container for such
1818 purposes. A relation is by definition a container for two @code{ex} and
1819 a relation between them that signals equality, inequality and so on.
1820 They are created by simply using the C++ operators @code{==}, @code{!=},
1821 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1823 @xref{Mathematical functions}, for examples where various applications
1824 of the @code{.subs()} method show how objects of class relational are
1825 used as arguments. There they provide an intuitive syntax for
1826 substitutions. They are also used as arguments to the @code{ex::series}
1827 method, where the left hand side of the relation specifies the variable
1828 to expand in and the right hand side the expansion point. They can also
1829 be used for creating systems of equations that are to be solved for
1830 unknown variables. But the most common usage of objects of this class
1831 is rather inconspicuous in statements of the form @code{if
1832 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1833 conversion from @code{relational} to @code{bool} takes place. Note,
1834 however, that @code{==} here does not perform any simplifications, hence
1835 @code{expand()} must be called explicitly.
1837 @node Integrals, Matrices, Relations, Basic Concepts
1838 @c node-name, next, previous, up
1840 @cindex @code{integral} (class)
1842 An object of class @dfn{integral} can be used to hold a symbolic integral.
1843 If you want to symbolically represent the integral of @code{x*x} from 0 to
1844 1, you would write this as
1846 integral(x, 0, 1, x*x)
1848 The first argument is the integration variable. It should be noted that
1849 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1850 fact, it can only integrate polynomials. An expression containing integrals
1851 can be evaluated symbolically by calling the
1855 method on it. Numerical evaluation is available by calling the
1859 method on an expression containing the integral. This will only evaluate
1860 integrals into a number if @code{subs}ing the integration variable by a
1861 number in the fourth argument of an integral and then @code{evalf}ing the
1862 result always results in a number. Of course, also the boundaries of the
1863 integration domain must @code{evalf} into numbers. It should be noted that
1864 trying to @code{evalf} a function with discontinuities in the integration
1865 domain is not recommended. The accuracy of the numeric evaluation of
1866 integrals is determined by the static member variable
1868 ex integral::relative_integration_error
1870 of the class @code{integral}. The default value of this is 10^-8.
1871 The integration works by halving the interval of integration, until numeric
1872 stability of the answer indicates that the requested accuracy has been
1873 reached. The maximum depth of the halving can be set via the static member
1876 int integral::max_integration_level
1878 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1879 return the integral unevaluated. The function that performs the numerical
1880 evaluation, is also available as
1882 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1885 This function will throw an exception if the maximum depth is exceeded. The
1886 last parameter of the function is optional and defaults to the
1887 @code{relative_integration_error}. To make sure that we do not do too
1888 much work if an expression contains the same integral multiple times,
1889 a lookup table is used.
1891 If you know that an expression holds an integral, you can get the
1892 integration variable, the left boundary, right boundary and integrant by
1893 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1894 @code{.op(3)}. Differentiating integrals with respect to variables works
1895 as expected. Note that it makes no sense to differentiate an integral
1896 with respect to the integration variable.
1898 @node Matrices, Indexed objects, Integrals, Basic Concepts
1899 @c node-name, next, previous, up
1901 @cindex @code{matrix} (class)
1903 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1904 matrix with @math{m} rows and @math{n} columns are accessed with two
1905 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1906 second one in the range 0@dots{}@math{n-1}.
1908 There are a couple of ways to construct matrices, with or without preset
1909 elements. The constructor
1912 matrix::matrix(unsigned r, unsigned c);
1915 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1918 The fastest way to create a matrix with preinitialized elements is to assign
1919 a list of comma-separated expressions to an empty matrix (see below for an
1920 example). But you can also specify the elements as a (flat) list with
1923 matrix::matrix(unsigned r, unsigned c, const lst & l);
1928 @cindex @code{lst_to_matrix()}
1930 ex lst_to_matrix(const lst & l);
1933 constructs a matrix from a list of lists, each list representing a matrix row.
1935 There is also a set of functions for creating some special types of
1938 @cindex @code{diag_matrix()}
1939 @cindex @code{unit_matrix()}
1940 @cindex @code{symbolic_matrix()}
1942 ex diag_matrix(const lst & l);
1943 ex unit_matrix(unsigned x);
1944 ex unit_matrix(unsigned r, unsigned c);
1945 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1946 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1949 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1950 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1951 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1952 matrix filled with newly generated symbols made of the specified base name
1953 and the position of each element in the matrix.
1955 Matrix elements can be accessed and set using the parenthesis (function call)
1959 const ex & matrix::operator()(unsigned r, unsigned c) const;
1960 ex & matrix::operator()(unsigned r, unsigned c);
1963 It is also possible to access the matrix elements in a linear fashion with
1964 the @code{op()} method. But C++-style subscripting with square brackets
1965 @samp{[]} is not available.
1967 Here are a couple of examples for constructing matrices:
1971 symbol a("a"), b("b");
1985 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1988 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1991 cout << diag_matrix(lst(a, b)) << endl;
1994 cout << unit_matrix(3) << endl;
1995 // -> [[1,0,0],[0,1,0],[0,0,1]]
1997 cout << symbolic_matrix(2, 3, "x") << endl;
1998 // -> [[x00,x01,x02],[x10,x11,x12]]
2002 @cindex @code{transpose()}
2003 There are three ways to do arithmetic with matrices. The first (and most
2004 direct one) is to use the methods provided by the @code{matrix} class:
2007 matrix matrix::add(const matrix & other) const;
2008 matrix matrix::sub(const matrix & other) const;
2009 matrix matrix::mul(const matrix & other) const;
2010 matrix matrix::mul_scalar(const ex & other) const;
2011 matrix matrix::pow(const ex & expn) const;
2012 matrix matrix::transpose() const;
2015 All of these methods return the result as a new matrix object. Here is an
2016 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2021 matrix A(2, 2), B(2, 2), C(2, 2);
2029 matrix result = A.mul(B).sub(C.mul_scalar(2));
2030 cout << result << endl;
2031 // -> [[-13,-6],[1,2]]
2036 @cindex @code{evalm()}
2037 The second (and probably the most natural) way is to construct an expression
2038 containing matrices with the usual arithmetic operators and @code{pow()}.
2039 For efficiency reasons, expressions with sums, products and powers of
2040 matrices are not automatically evaluated in GiNaC. You have to call the
2044 ex ex::evalm() const;
2047 to obtain the result:
2054 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2055 cout << e.evalm() << endl;
2056 // -> [[-13,-6],[1,2]]
2061 The non-commutativity of the product @code{A*B} in this example is
2062 automatically recognized by GiNaC. There is no need to use a special
2063 operator here. @xref{Non-commutative objects}, for more information about
2064 dealing with non-commutative expressions.
2066 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2067 to perform the arithmetic:
2072 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2073 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2075 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2076 cout << e.simplify_indexed() << endl;
2077 // -> [[-13,-6],[1,2]].i.j
2081 Using indices is most useful when working with rectangular matrices and
2082 one-dimensional vectors because you don't have to worry about having to
2083 transpose matrices before multiplying them. @xref{Indexed objects}, for
2084 more information about using matrices with indices, and about indices in
2087 The @code{matrix} class provides a couple of additional methods for
2088 computing determinants, traces, characteristic polynomials and ranks:
2090 @cindex @code{determinant()}
2091 @cindex @code{trace()}
2092 @cindex @code{charpoly()}
2093 @cindex @code{rank()}
2095 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2096 ex matrix::trace() const;
2097 ex matrix::charpoly(const ex & lambda) const;
2098 unsigned matrix::rank() const;
2101 The @samp{algo} argument of @code{determinant()} allows to select
2102 between different algorithms for calculating the determinant. The
2103 asymptotic speed (as parametrized by the matrix size) can greatly differ
2104 between those algorithms, depending on the nature of the matrix'
2105 entries. The possible values are defined in the @file{flags.h} header
2106 file. By default, GiNaC uses a heuristic to automatically select an
2107 algorithm that is likely (but not guaranteed) to give the result most
2110 @cindex @code{inverse()} (matrix)
2111 @cindex @code{solve()}
2112 Matrices may also be inverted using the @code{ex matrix::inverse()}
2113 method and linear systems may be solved with:
2116 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
2119 Assuming the matrix object this method is applied on is an @code{m}
2120 times @code{n} matrix, then @code{vars} must be a @code{n} times
2121 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2122 times @code{p} matrix. The returned matrix then has dimension @code{n}
2123 times @code{p} and in the case of an underdetermined system will still
2124 contain some of the indeterminates from @code{vars}. If the system is
2125 overdetermined, an exception is thrown.
2128 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2129 @c node-name, next, previous, up
2130 @section Indexed objects
2132 GiNaC allows you to handle expressions containing general indexed objects in
2133 arbitrary spaces. It is also able to canonicalize and simplify such
2134 expressions and perform symbolic dummy index summations. There are a number
2135 of predefined indexed objects provided, like delta and metric tensors.
2137 There are few restrictions placed on indexed objects and their indices and
2138 it is easy to construct nonsense expressions, but our intention is to
2139 provide a general framework that allows you to implement algorithms with
2140 indexed quantities, getting in the way as little as possible.
2142 @cindex @code{idx} (class)
2143 @cindex @code{indexed} (class)
2144 @subsection Indexed quantities and their indices
2146 Indexed expressions in GiNaC are constructed of two special types of objects,
2147 @dfn{index objects} and @dfn{indexed objects}.
2151 @cindex contravariant
2154 @item Index objects are of class @code{idx} or a subclass. Every index has
2155 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2156 the index lives in) which can both be arbitrary expressions but are usually
2157 a number or a simple symbol. In addition, indices of class @code{varidx} have
2158 a @dfn{variance} (they can be co- or contravariant), and indices of class
2159 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2161 @item Indexed objects are of class @code{indexed} or a subclass. They
2162 contain a @dfn{base expression} (which is the expression being indexed), and
2163 one or more indices.
2167 @strong{Note:} when printing expressions, covariant indices and indices
2168 without variance are denoted @samp{.i} while contravariant indices are
2169 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2170 value. In the following, we are going to use that notation in the text so
2171 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2172 not visible in the output.
2174 A simple example shall illustrate the concepts:
2178 #include <ginac/ginac.h>
2179 using namespace std;
2180 using namespace GiNaC;
2184 symbol i_sym("i"), j_sym("j");
2185 idx i(i_sym, 3), j(j_sym, 3);
2188 cout << indexed(A, i, j) << endl;
2190 cout << index_dimensions << indexed(A, i, j) << endl;
2192 cout << dflt; // reset cout to default output format (dimensions hidden)
2196 The @code{idx} constructor takes two arguments, the index value and the
2197 index dimension. First we define two index objects, @code{i} and @code{j},
2198 both with the numeric dimension 3. The value of the index @code{i} is the
2199 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2200 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2201 construct an expression containing one indexed object, @samp{A.i.j}. It has
2202 the symbol @code{A} as its base expression and the two indices @code{i} and
2205 The dimensions of indices are normally not visible in the output, but one
2206 can request them to be printed with the @code{index_dimensions} manipulator,
2209 Note the difference between the indices @code{i} and @code{j} which are of
2210 class @code{idx}, and the index values which are the symbols @code{i_sym}
2211 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2212 or numbers but must be index objects. For example, the following is not
2213 correct and will raise an exception:
2216 symbol i("i"), j("j");
2217 e = indexed(A, i, j); // ERROR: indices must be of type idx
2220 You can have multiple indexed objects in an expression, index values can
2221 be numeric, and index dimensions symbolic:
2225 symbol B("B"), dim("dim");
2226 cout << 4 * indexed(A, i)
2227 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2232 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2233 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2234 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2235 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2236 @code{simplify_indexed()} for that, see below).
2238 In fact, base expressions, index values and index dimensions can be
2239 arbitrary expressions:
2243 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2248 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2249 get an error message from this but you will probably not be able to do
2250 anything useful with it.
2252 @cindex @code{get_value()}
2253 @cindex @code{get_dimension()}
2257 ex idx::get_value();
2258 ex idx::get_dimension();
2261 return the value and dimension of an @code{idx} object. If you have an index
2262 in an expression, such as returned by calling @code{.op()} on an indexed
2263 object, you can get a reference to the @code{idx} object with the function
2264 @code{ex_to<idx>()} on the expression.
2266 There are also the methods
2269 bool idx::is_numeric();
2270 bool idx::is_symbolic();
2271 bool idx::is_dim_numeric();
2272 bool idx::is_dim_symbolic();
2275 for checking whether the value and dimension are numeric or symbolic
2276 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2277 About Expressions}) returns information about the index value.
2279 @cindex @code{varidx} (class)
2280 If you need co- and contravariant indices, use the @code{varidx} class:
2284 symbol mu_sym("mu"), nu_sym("nu");
2285 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2286 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2288 cout << indexed(A, mu, nu) << endl;
2290 cout << indexed(A, mu_co, nu) << endl;
2292 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2297 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2298 co- or contravariant. The default is a contravariant (upper) index, but
2299 this can be overridden by supplying a third argument to the @code{varidx}
2300 constructor. The two methods
2303 bool varidx::is_covariant();
2304 bool varidx::is_contravariant();
2307 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2308 to get the object reference from an expression). There's also the very useful
2312 ex varidx::toggle_variance();
2315 which makes a new index with the same value and dimension but the opposite
2316 variance. By using it you only have to define the index once.
2318 @cindex @code{spinidx} (class)
2319 The @code{spinidx} class provides dotted and undotted variant indices, as
2320 used in the Weyl-van-der-Waerden spinor formalism:
2324 symbol K("K"), C_sym("C"), D_sym("D");
2325 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2326 // contravariant, undotted
2327 spinidx C_co(C_sym, 2, true); // covariant index
2328 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2329 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2331 cout << indexed(K, C, D) << endl;
2333 cout << indexed(K, C_co, D_dot) << endl;
2335 cout << indexed(K, D_co_dot, D) << endl;
2340 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2341 dotted or undotted. The default is undotted but this can be overridden by
2342 supplying a fourth argument to the @code{spinidx} constructor. The two
2346 bool spinidx::is_dotted();
2347 bool spinidx::is_undotted();
2350 allow you to check whether or not a @code{spinidx} object is dotted (use
2351 @code{ex_to<spinidx>()} to get the object reference from an expression).
2352 Finally, the two methods
2355 ex spinidx::toggle_dot();
2356 ex spinidx::toggle_variance_dot();
2359 create a new index with the same value and dimension but opposite dottedness
2360 and the same or opposite variance.
2362 @subsection Substituting indices
2364 @cindex @code{subs()}
2365 Sometimes you will want to substitute one symbolic index with another
2366 symbolic or numeric index, for example when calculating one specific element
2367 of a tensor expression. This is done with the @code{.subs()} method, as it
2368 is done for symbols (see @ref{Substituting Expressions}).
2370 You have two possibilities here. You can either substitute the whole index
2371 by another index or expression:
2375 ex e = indexed(A, mu_co);
2376 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2377 // -> A.mu becomes A~nu
2378 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2379 // -> A.mu becomes A~0
2380 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2381 // -> A.mu becomes A.0
2385 The third example shows that trying to replace an index with something that
2386 is not an index will substitute the index value instead.
2388 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2393 ex e = indexed(A, mu_co);
2394 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2395 // -> A.mu becomes A.nu
2396 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2397 // -> A.mu becomes A.0
2401 As you see, with the second method only the value of the index will get
2402 substituted. Its other properties, including its dimension, remain unchanged.
2403 If you want to change the dimension of an index you have to substitute the
2404 whole index by another one with the new dimension.
2406 Finally, substituting the base expression of an indexed object works as
2411 ex e = indexed(A, mu_co);
2412 cout << e << " becomes " << e.subs(A == A+B) << endl;
2413 // -> A.mu becomes (B+A).mu
2417 @subsection Symmetries
2418 @cindex @code{symmetry} (class)
2419 @cindex @code{sy_none()}
2420 @cindex @code{sy_symm()}
2421 @cindex @code{sy_anti()}
2422 @cindex @code{sy_cycl()}
2424 Indexed objects can have certain symmetry properties with respect to their
2425 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2426 that is constructed with the helper functions
2429 symmetry sy_none(...);
2430 symmetry sy_symm(...);
2431 symmetry sy_anti(...);
2432 symmetry sy_cycl(...);
2435 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2436 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2437 represents a cyclic symmetry. Each of these functions accepts up to four
2438 arguments which can be either symmetry objects themselves or unsigned integer
2439 numbers that represent an index position (counting from 0). A symmetry
2440 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2441 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2444 Here are some examples of symmetry definitions:
2449 e = indexed(A, i, j);
2450 e = indexed(A, sy_none(), i, j); // equivalent
2451 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2453 // Symmetric in all three indices:
2454 e = indexed(A, sy_symm(), i, j, k);
2455 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2456 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2457 // different canonical order
2459 // Symmetric in the first two indices only:
2460 e = indexed(A, sy_symm(0, 1), i, j, k);
2461 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2463 // Antisymmetric in the first and last index only (index ranges need not
2465 e = indexed(A, sy_anti(0, 2), i, j, k);
2466 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2468 // An example of a mixed symmetry: antisymmetric in the first two and
2469 // last two indices, symmetric when swapping the first and last index
2470 // pairs (like the Riemann curvature tensor):
2471 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2473 // Cyclic symmetry in all three indices:
2474 e = indexed(A, sy_cycl(), i, j, k);
2475 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2477 // The following examples are invalid constructions that will throw
2478 // an exception at run time.
2480 // An index may not appear multiple times:
2481 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2482 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2484 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2485 // same number of indices:
2486 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2488 // And of course, you cannot specify indices which are not there:
2489 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2493 If you need to specify more than four indices, you have to use the
2494 @code{.add()} method of the @code{symmetry} class. For example, to specify
2495 full symmetry in the first six indices you would write
2496 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2498 If an indexed object has a symmetry, GiNaC will automatically bring the
2499 indices into a canonical order which allows for some immediate simplifications:
2503 cout << indexed(A, sy_symm(), i, j)
2504 + indexed(A, sy_symm(), j, i) << endl;
2506 cout << indexed(B, sy_anti(), i, j)
2507 + indexed(B, sy_anti(), j, i) << endl;
2509 cout << indexed(B, sy_anti(), i, j, k)
2510 - indexed(B, sy_anti(), j, k, i) << endl;
2515 @cindex @code{get_free_indices()}
2517 @subsection Dummy indices
2519 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2520 that a summation over the index range is implied. Symbolic indices which are
2521 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2522 dummy nor free indices.
2524 To be recognized as a dummy index pair, the two indices must be of the same
2525 class and their value must be the same single symbol (an index like
2526 @samp{2*n+1} is never a dummy index). If the indices are of class
2527 @code{varidx} they must also be of opposite variance; if they are of class
2528 @code{spinidx} they must be both dotted or both undotted.
2530 The method @code{.get_free_indices()} returns a vector containing the free
2531 indices of an expression. It also checks that the free indices of the terms
2532 of a sum are consistent:
2536 symbol A("A"), B("B"), C("C");
2538 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2539 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2541 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2542 cout << exprseq(e.get_free_indices()) << endl;
2544 // 'j' and 'l' are dummy indices
2546 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2547 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2549 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2550 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2551 cout << exprseq(e.get_free_indices()) << endl;
2553 // 'nu' is a dummy index, but 'sigma' is not
2555 e = indexed(A, mu, mu);
2556 cout << exprseq(e.get_free_indices()) << endl;
2558 // 'mu' is not a dummy index because it appears twice with the same
2561 e = indexed(A, mu, nu) + 42;
2562 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2563 // this will throw an exception:
2564 // "add::get_free_indices: inconsistent indices in sum"
2568 @cindex @code{simplify_indexed()}
2569 @subsection Simplifying indexed expressions
2571 In addition to the few automatic simplifications that GiNaC performs on
2572 indexed expressions (such as re-ordering the indices of symmetric tensors
2573 and calculating traces and convolutions of matrices and predefined tensors)
2577 ex ex::simplify_indexed();
2578 ex ex::simplify_indexed(const scalar_products & sp);
2581 that performs some more expensive operations:
2584 @item it checks the consistency of free indices in sums in the same way
2585 @code{get_free_indices()} does
2586 @item it tries to give dummy indices that appear in different terms of a sum
2587 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2588 @item it (symbolically) calculates all possible dummy index summations/contractions
2589 with the predefined tensors (this will be explained in more detail in the
2591 @item it detects contractions that vanish for symmetry reasons, for example
2592 the contraction of a symmetric and a totally antisymmetric tensor
2593 @item as a special case of dummy index summation, it can replace scalar products
2594 of two tensors with a user-defined value
2597 The last point is done with the help of the @code{scalar_products} class
2598 which is used to store scalar products with known values (this is not an
2599 arithmetic class, you just pass it to @code{simplify_indexed()}):
2603 symbol A("A"), B("B"), C("C"), i_sym("i");
2607 sp.add(A, B, 0); // A and B are orthogonal
2608 sp.add(A, C, 0); // A and C are orthogonal
2609 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2611 e = indexed(A + B, i) * indexed(A + C, i);
2613 // -> (B+A).i*(A+C).i
2615 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2621 The @code{scalar_products} object @code{sp} acts as a storage for the
2622 scalar products added to it with the @code{.add()} method. This method
2623 takes three arguments: the two expressions of which the scalar product is
2624 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2625 @code{simplify_indexed()} will replace all scalar products of indexed
2626 objects that have the symbols @code{A} and @code{B} as base expressions
2627 with the single value 0. The number, type and dimension of the indices
2628 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2630 @cindex @code{expand()}
2631 The example above also illustrates a feature of the @code{expand()} method:
2632 if passed the @code{expand_indexed} option it will distribute indices
2633 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2635 @cindex @code{tensor} (class)
2636 @subsection Predefined tensors
2638 Some frequently used special tensors such as the delta, epsilon and metric
2639 tensors are predefined in GiNaC. They have special properties when
2640 contracted with other tensor expressions and some of them have constant
2641 matrix representations (they will evaluate to a number when numeric
2642 indices are specified).
2644 @cindex @code{delta_tensor()}
2645 @subsubsection Delta tensor
2647 The delta tensor takes two indices, is symmetric and has the matrix
2648 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2649 @code{delta_tensor()}:
2653 symbol A("A"), B("B");
2655 idx i(symbol("i"), 3), j(symbol("j"), 3),
2656 k(symbol("k"), 3), l(symbol("l"), 3);
2658 ex e = indexed(A, i, j) * indexed(B, k, l)
2659 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2660 cout << e.simplify_indexed() << endl;
2663 cout << delta_tensor(i, i) << endl;
2668 @cindex @code{metric_tensor()}
2669 @subsubsection General metric tensor
2671 The function @code{metric_tensor()} creates a general symmetric metric
2672 tensor with two indices that can be used to raise/lower tensor indices. The
2673 metric tensor is denoted as @samp{g} in the output and if its indices are of
2674 mixed variance it is automatically replaced by a delta tensor:
2680 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2682 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2683 cout << e.simplify_indexed() << endl;
2686 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2687 cout << e.simplify_indexed() << endl;
2690 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2691 * metric_tensor(nu, rho);
2692 cout << e.simplify_indexed() << endl;
2695 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2696 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2697 + indexed(A, mu.toggle_variance(), rho));
2698 cout << e.simplify_indexed() << endl;
2703 @cindex @code{lorentz_g()}
2704 @subsubsection Minkowski metric tensor
2706 The Minkowski metric tensor is a special metric tensor with a constant
2707 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2708 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2709 It is created with the function @code{lorentz_g()} (although it is output as
2714 varidx mu(symbol("mu"), 4);
2716 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2717 * lorentz_g(mu, varidx(0, 4)); // negative signature
2718 cout << e.simplify_indexed() << endl;
2721 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2722 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2723 cout << e.simplify_indexed() << endl;
2728 @cindex @code{spinor_metric()}
2729 @subsubsection Spinor metric tensor
2731 The function @code{spinor_metric()} creates an antisymmetric tensor with
2732 two indices that is used to raise/lower indices of 2-component spinors.
2733 It is output as @samp{eps}:
2739 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2740 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2742 e = spinor_metric(A, B) * indexed(psi, B_co);
2743 cout << e.simplify_indexed() << endl;
2746 e = spinor_metric(A, B) * indexed(psi, A_co);
2747 cout << e.simplify_indexed() << endl;
2750 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2751 cout << e.simplify_indexed() << endl;
2754 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2755 cout << e.simplify_indexed() << endl;
2758 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2759 cout << e.simplify_indexed() << endl;
2762 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2763 cout << e.simplify_indexed() << endl;
2768 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2770 @cindex @code{epsilon_tensor()}
2771 @cindex @code{lorentz_eps()}
2772 @subsubsection Epsilon tensor
2774 The epsilon tensor is totally antisymmetric, its number of indices is equal
2775 to the dimension of the index space (the indices must all be of the same
2776 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2777 defined to be 1. Its behavior with indices that have a variance also
2778 depends on the signature of the metric. Epsilon tensors are output as
2781 There are three functions defined to create epsilon tensors in 2, 3 and 4
2785 ex epsilon_tensor(const ex & i1, const ex & i2);
2786 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2787 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2790 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2791 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2792 Minkowski space (the last @code{bool} argument specifies whether the metric
2793 has negative or positive signature, as in the case of the Minkowski metric
2798 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2799 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2800 e = lorentz_eps(mu, nu, rho, sig) *
2801 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2802 cout << simplify_indexed(e) << endl;
2803 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2805 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2806 symbol A("A"), B("B");
2807 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2808 cout << simplify_indexed(e) << endl;
2809 // -> -B.k*A.j*eps.i.k.j
2810 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2811 cout << simplify_indexed(e) << endl;
2816 @subsection Linear algebra
2818 The @code{matrix} class can be used with indices to do some simple linear
2819 algebra (linear combinations and products of vectors and matrices, traces
2820 and scalar products):
2824 idx i(symbol("i"), 2), j(symbol("j"), 2);
2825 symbol x("x"), y("y");
2827 // A is a 2x2 matrix, X is a 2x1 vector
2828 matrix A(2, 2), X(2, 1);
2833 cout << indexed(A, i, i) << endl;
2836 ex e = indexed(A, i, j) * indexed(X, j);
2837 cout << e.simplify_indexed() << endl;
2838 // -> [[2*y+x],[4*y+3*x]].i
2840 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2841 cout << e.simplify_indexed() << endl;
2842 // -> [[3*y+3*x,6*y+2*x]].j
2846 You can of course obtain the same results with the @code{matrix::add()},
2847 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2848 but with indices you don't have to worry about transposing matrices.
2850 Matrix indices always start at 0 and their dimension must match the number
2851 of rows/columns of the matrix. Matrices with one row or one column are
2852 vectors and can have one or two indices (it doesn't matter whether it's a
2853 row or a column vector). Other matrices must have two indices.
2855 You should be careful when using indices with variance on matrices. GiNaC
2856 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2857 @samp{F.mu.nu} are different matrices. In this case you should use only
2858 one form for @samp{F} and explicitly multiply it with a matrix representation
2859 of the metric tensor.
2862 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2863 @c node-name, next, previous, up
2864 @section Non-commutative objects
2866 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2867 non-commutative objects are built-in which are mostly of use in high energy
2871 @item Clifford (Dirac) algebra (class @code{clifford})
2872 @item su(3) Lie algebra (class @code{color})
2873 @item Matrices (unindexed) (class @code{matrix})
2876 The @code{clifford} and @code{color} classes are subclasses of
2877 @code{indexed} because the elements of these algebras usually carry
2878 indices. The @code{matrix} class is described in more detail in
2881 Unlike most computer algebra systems, GiNaC does not primarily provide an
2882 operator (often denoted @samp{&*}) for representing inert products of
2883 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2884 classes of objects involved, and non-commutative products are formed with
2885 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2886 figuring out by itself which objects commutate and will group the factors
2887 by their class. Consider this example:
2891 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2892 idx a(symbol("a"), 8), b(symbol("b"), 8);
2893 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2895 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2899 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2900 groups the non-commutative factors (the gammas and the su(3) generators)
2901 together while preserving the order of factors within each class (because
2902 Clifford objects commutate with color objects). The resulting expression is a
2903 @emph{commutative} product with two factors that are themselves non-commutative
2904 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2905 parentheses are placed around the non-commutative products in the output.
2907 @cindex @code{ncmul} (class)
2908 Non-commutative products are internally represented by objects of the class
2909 @code{ncmul}, as opposed to commutative products which are handled by the
2910 @code{mul} class. You will normally not have to worry about this distinction,
2913 The advantage of this approach is that you never have to worry about using
2914 (or forgetting to use) a special operator when constructing non-commutative
2915 expressions. Also, non-commutative products in GiNaC are more intelligent
2916 than in other computer algebra systems; they can, for example, automatically
2917 canonicalize themselves according to rules specified in the implementation
2918 of the non-commutative classes. The drawback is that to work with other than
2919 the built-in algebras you have to implement new classes yourself. Symbols
2920 always commutate and it's not possible to construct non-commutative products
2921 using symbols to represent the algebra elements or generators. User-defined
2922 functions can, however, be specified as being non-commutative.
2924 @cindex @code{return_type()}
2925 @cindex @code{return_type_tinfo()}
2926 Information about the commutativity of an object or expression can be
2927 obtained with the two member functions
2930 unsigned ex::return_type() const;
2931 unsigned ex::return_type_tinfo() const;
2934 The @code{return_type()} function returns one of three values (defined in
2935 the header file @file{flags.h}), corresponding to three categories of
2936 expressions in GiNaC:
2939 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2940 classes are of this kind.
2941 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2942 certain class of non-commutative objects which can be determined with the
2943 @code{return_type_tinfo()} method. Expressions of this category commutate
2944 with everything except @code{noncommutative} expressions of the same
2946 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2947 of non-commutative objects of different classes. Expressions of this
2948 category don't commutate with any other @code{noncommutative} or
2949 @code{noncommutative_composite} expressions.
2952 The value returned by the @code{return_type_tinfo()} method is valid only
2953 when the return type of the expression is @code{noncommutative}. It is a
2954 value that is unique to the class of the object and usually one of the
2955 constants in @file{tinfos.h}, or derived therefrom.
2957 Here are a couple of examples:
2960 @multitable @columnfractions 0.33 0.33 0.34
2961 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2962 @item @code{42} @tab @code{commutative} @tab -
2963 @item @code{2*x-y} @tab @code{commutative} @tab -
2964 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2965 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2966 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2967 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2971 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2972 @code{TINFO_clifford} for objects with a representation label of zero.
2973 Other representation labels yield a different @code{return_type_tinfo()},
2974 but it's the same for any two objects with the same label. This is also true
2977 A last note: With the exception of matrices, positive integer powers of
2978 non-commutative objects are automatically expanded in GiNaC. For example,
2979 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2980 non-commutative expressions).
2983 @cindex @code{clifford} (class)
2984 @subsection Clifford algebra
2986 @cindex @code{dirac_gamma()}
2987 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2988 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2989 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2990 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2993 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2996 which takes two arguments: the index and a @dfn{representation label} in the
2997 range 0 to 255 which is used to distinguish elements of different Clifford
2998 algebras (this is also called a @dfn{spin line index}). Gammas with different
2999 labels commutate with each other. The dimension of the index can be 4 or (in
3000 the framework of dimensional regularization) any symbolic value. Spinor
3001 indices on Dirac gammas are not supported in GiNaC.
3003 @cindex @code{dirac_ONE()}
3004 The unity element of a Clifford algebra is constructed by
3007 ex dirac_ONE(unsigned char rl = 0);
3010 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
3011 multiples of the unity element, even though it's customary to omit it.
3012 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3013 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3014 GiNaC will complain and/or produce incorrect results.
3016 @cindex @code{dirac_gamma5()}
3017 There is a special element @samp{gamma5} that commutates with all other
3018 gammas, has a unit square, and in 4 dimensions equals
3019 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3022 ex dirac_gamma5(unsigned char rl = 0);
3025 @cindex @code{dirac_gammaL()}
3026 @cindex @code{dirac_gammaR()}
3027 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3028 objects, constructed by
3031 ex dirac_gammaL(unsigned char rl = 0);
3032 ex dirac_gammaR(unsigned char rl = 0);
3035 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3036 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3038 @cindex @code{dirac_slash()}
3039 Finally, the function
3042 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3045 creates a term that represents a contraction of @samp{e} with the Dirac
3046 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3047 with a unique index whose dimension is given by the @code{dim} argument).
3048 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3050 In products of dirac gammas, superfluous unity elements are automatically
3051 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3052 and @samp{gammaR} are moved to the front.
3054 The @code{simplify_indexed()} function performs contractions in gamma strings,
3060 symbol a("a"), b("b"), D("D");
3061 varidx mu(symbol("mu"), D);
3062 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3063 * dirac_gamma(mu.toggle_variance());
3065 // -> gamma~mu*a\*gamma.mu
3066 e = e.simplify_indexed();
3069 cout << e.subs(D == 4) << endl;
3075 @cindex @code{dirac_trace()}
3076 To calculate the trace of an expression containing strings of Dirac gammas
3077 you use one of the functions
3080 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
3081 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3082 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3085 These functions take the trace over all gammas in the specified set @code{rls}
3086 or list @code{rll} of representation labels, or the single label @code{rl};
3087 gammas with other labels are left standing. The last argument to
3088 @code{dirac_trace()} is the value to be returned for the trace of the unity
3089 element, which defaults to 4.
3091 The @code{dirac_trace()} function is a linear functional that is equal to the
3092 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3093 functional is not cyclic in @math{D != 4} dimensions when acting on
3094 expressions containing @samp{gamma5}, so it's not a proper trace. This
3095 @samp{gamma5} scheme is described in greater detail in
3096 @cite{The Role of gamma5 in Dimensional Regularization}.
3098 The value of the trace itself is also usually different in 4 and in
3099 @math{D != 4} dimensions:
3104 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3105 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3106 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3107 cout << dirac_trace(e).simplify_indexed() << endl;
3114 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3115 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3116 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3117 cout << dirac_trace(e).simplify_indexed() << endl;
3118 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3122 Here is an example for using @code{dirac_trace()} to compute a value that
3123 appears in the calculation of the one-loop vacuum polarization amplitude in
3128 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3129 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3132 sp.add(l, l, pow(l, 2));
3133 sp.add(l, q, ldotq);
3135 ex e = dirac_gamma(mu) *
3136 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3137 dirac_gamma(mu.toggle_variance()) *
3138 (dirac_slash(l, D) + m * dirac_ONE());
3139 e = dirac_trace(e).simplify_indexed(sp);
3140 e = e.collect(lst(l, ldotq, m));
3142 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3146 The @code{canonicalize_clifford()} function reorders all gamma products that
3147 appear in an expression to a canonical (but not necessarily simple) form.
3148 You can use this to compare two expressions or for further simplifications:
3152 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3153 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3155 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3157 e = canonicalize_clifford(e);
3159 // -> 2*ONE*eta~mu~nu
3164 @cindex @code{color} (class)
3165 @subsection Color algebra
3167 @cindex @code{color_T()}
3168 For computations in quantum chromodynamics, GiNaC implements the base elements
3169 and structure constants of the su(3) Lie algebra (color algebra). The base
3170 elements @math{T_a} are constructed by the function
3173 ex color_T(const ex & a, unsigned char rl = 0);
3176 which takes two arguments: the index and a @dfn{representation label} in the
3177 range 0 to 255 which is used to distinguish elements of different color
3178 algebras. Objects with different labels commutate with each other. The
3179 dimension of the index must be exactly 8 and it should be of class @code{idx},
3182 @cindex @code{color_ONE()}
3183 The unity element of a color algebra is constructed by
3186 ex color_ONE(unsigned char rl = 0);
3189 @strong{Note:} You must always use @code{color_ONE()} when referring to
3190 multiples of the unity element, even though it's customary to omit it.
3191 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3192 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3193 GiNaC may produce incorrect results.
3195 @cindex @code{color_d()}
3196 @cindex @code{color_f()}
3200 ex color_d(const ex & a, const ex & b, const ex & c);
3201 ex color_f(const ex & a, const ex & b, const ex & c);
3204 create the symmetric and antisymmetric structure constants @math{d_abc} and
3205 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3206 and @math{[T_a, T_b] = i f_abc T_c}.
3208 @cindex @code{color_h()}
3209 There's an additional function
3212 ex color_h(const ex & a, const ex & b, const ex & c);
3215 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3217 The function @code{simplify_indexed()} performs some simplifications on
3218 expressions containing color objects:
3223 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3224 k(symbol("k"), 8), l(symbol("l"), 8);
3226 e = color_d(a, b, l) * color_f(a, b, k);
3227 cout << e.simplify_indexed() << endl;
3230 e = color_d(a, b, l) * color_d(a, b, k);
3231 cout << e.simplify_indexed() << endl;
3234 e = color_f(l, a, b) * color_f(a, b, k);
3235 cout << e.simplify_indexed() << endl;
3238 e = color_h(a, b, c) * color_h(a, b, c);
3239 cout << e.simplify_indexed() << endl;
3242 e = color_h(a, b, c) * color_T(b) * color_T(c);
3243 cout << e.simplify_indexed() << endl;
3246 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3247 cout << e.simplify_indexed() << endl;
3250 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3251 cout << e.simplify_indexed() << endl;
3252 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3256 @cindex @code{color_trace()}
3257 To calculate the trace of an expression containing color objects you use one
3261 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3262 ex color_trace(const ex & e, const lst & rll);
3263 ex color_trace(const ex & e, unsigned char rl = 0);
3266 These functions take the trace over all color @samp{T} objects in the
3267 specified set @code{rls} or list @code{rll} of representation labels, or the
3268 single label @code{rl}; @samp{T}s with other labels are left standing. For
3273 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3275 // -> -I*f.a.c.b+d.a.c.b
3280 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3281 @c node-name, next, previous, up
3284 @cindex @code{exhashmap} (class)
3286 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3287 that can be used as a drop-in replacement for the STL
3288 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3289 typically constant-time, element look-up than @code{map<>}.
3291 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3292 following differences:
3296 no @code{lower_bound()} and @code{upper_bound()} methods
3298 no reverse iterators, no @code{rbegin()}/@code{rend()}
3300 no @code{operator<(exhashmap, exhashmap)}
3302 the comparison function object @code{key_compare} is hardcoded to
3305 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3306 initial hash table size (the actual table size after construction may be
3307 larger than the specified value)
3309 the method @code{size_t bucket_count()} returns the current size of the hash
3312 @code{insert()} and @code{erase()} operations invalidate all iterators
3316 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3317 @c node-name, next, previous, up
3318 @chapter Methods and Functions
3321 In this chapter the most important algorithms provided by GiNaC will be
3322 described. Some of them are implemented as functions on expressions,
3323 others are implemented as methods provided by expression objects. If
3324 they are methods, there exists a wrapper function around it, so you can
3325 alternatively call it in a functional way as shown in the simple
3330 cout << "As method: " << sin(1).evalf() << endl;
3331 cout << "As function: " << evalf(sin(1)) << endl;
3335 @cindex @code{subs()}
3336 The general rule is that wherever methods accept one or more parameters
3337 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3338 wrapper accepts is the same but preceded by the object to act on
3339 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3340 most natural one in an OO model but it may lead to confusion for MapleV
3341 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3342 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3343 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3344 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3345 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3346 here. Also, users of MuPAD will in most cases feel more comfortable
3347 with GiNaC's convention. All function wrappers are implemented
3348 as simple inline functions which just call the corresponding method and
3349 are only provided for users uncomfortable with OO who are dead set to
3350 avoid method invocations. Generally, nested function wrappers are much
3351 harder to read than a sequence of methods and should therefore be
3352 avoided if possible. On the other hand, not everything in GiNaC is a
3353 method on class @code{ex} and sometimes calling a function cannot be
3357 * Information About Expressions::
3358 * Numerical Evaluation::
3359 * Substituting Expressions::
3360 * Pattern Matching and Advanced Substitutions::
3361 * Applying a Function on Subexpressions::
3362 * Visitors and Tree Traversal::
3363 * Polynomial Arithmetic:: Working with polynomials.
3364 * Rational Expressions:: Working with rational functions.
3365 * Symbolic Differentiation::
3366 * Series Expansion:: Taylor and Laurent expansion.
3368 * Built-in Functions:: List of predefined mathematical functions.
3369 * Multiple polylogarithms::
3370 * Complex Conjugation::
3371 * Built-in Functions:: List of predefined mathematical functions.
3372 * Solving Linear Systems of Equations::
3373 * Input/Output:: Input and output of expressions.
3377 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3378 @c node-name, next, previous, up
3379 @section Getting information about expressions
3381 @subsection Checking expression types
3382 @cindex @code{is_a<@dots{}>()}
3383 @cindex @code{is_exactly_a<@dots{}>()}
3384 @cindex @code{ex_to<@dots{}>()}
3385 @cindex Converting @code{ex} to other classes
3386 @cindex @code{info()}
3387 @cindex @code{return_type()}
3388 @cindex @code{return_type_tinfo()}
3390 Sometimes it's useful to check whether a given expression is a plain number,
3391 a sum, a polynomial with integer coefficients, or of some other specific type.
3392 GiNaC provides a couple of functions for this:
3395 bool is_a<T>(const ex & e);
3396 bool is_exactly_a<T>(const ex & e);
3397 bool ex::info(unsigned flag);
3398 unsigned ex::return_type() const;
3399 unsigned ex::return_type_tinfo() const;
3402 When the test made by @code{is_a<T>()} returns true, it is safe to call
3403 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3404 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3405 example, assuming @code{e} is an @code{ex}:
3410 if (is_a<numeric>(e))
3411 numeric n = ex_to<numeric>(e);
3416 @code{is_a<T>(e)} allows you to check whether the top-level object of
3417 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3418 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3419 e.g., for checking whether an expression is a number, a sum, or a product:
3426 is_a<numeric>(e1); // true
3427 is_a<numeric>(e2); // false
3428 is_a<add>(e1); // false
3429 is_a<add>(e2); // true
3430 is_a<mul>(e1); // false
3431 is_a<mul>(e2); // false
3435 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3436 top-level object of an expression @samp{e} is an instance of the GiNaC
3437 class @samp{T}, not including parent classes.
3439 The @code{info()} method is used for checking certain attributes of
3440 expressions. The possible values for the @code{flag} argument are defined
3441 in @file{ginac/flags.h}, the most important being explained in the following
3445 @multitable @columnfractions .30 .70
3446 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3447 @item @code{numeric}
3448 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3450 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3451 @item @code{rational}
3452 @tab @dots{}an exact rational number (integers are rational, too)
3453 @item @code{integer}
3454 @tab @dots{}a (non-complex) integer
3455 @item @code{crational}
3456 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3457 @item @code{cinteger}
3458 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3459 @item @code{positive}
3460 @tab @dots{}not complex and greater than 0
3461 @item @code{negative}
3462 @tab @dots{}not complex and less than 0
3463 @item @code{nonnegative}
3464 @tab @dots{}not complex and greater than or equal to 0
3466 @tab @dots{}an integer greater than 0
3468 @tab @dots{}an integer less than 0
3469 @item @code{nonnegint}
3470 @tab @dots{}an integer greater than or equal to 0
3472 @tab @dots{}an even integer
3474 @tab @dots{}an odd integer
3476 @tab @dots{}a prime integer (probabilistic primality test)
3477 @item @code{relation}
3478 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3479 @item @code{relation_equal}
3480 @tab @dots{}a @code{==} relation
3481 @item @code{relation_not_equal}
3482 @tab @dots{}a @code{!=} relation
3483 @item @code{relation_less}
3484 @tab @dots{}a @code{<} relation
3485 @item @code{relation_less_or_equal}
3486 @tab @dots{}a @code{<=} relation
3487 @item @code{relation_greater}
3488 @tab @dots{}a @code{>} relation
3489 @item @code{relation_greater_or_equal}
3490 @tab @dots{}a @code{>=} relation
3492 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3494 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3495 @item @code{polynomial}
3496 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3497 @item @code{integer_polynomial}
3498 @tab @dots{}a polynomial with (non-complex) integer coefficients
3499 @item @code{cinteger_polynomial}
3500 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3501 @item @code{rational_polynomial}
3502 @tab @dots{}a polynomial with (non-complex) rational coefficients
3503 @item @code{crational_polynomial}
3504 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3505 @item @code{rational_function}
3506 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3507 @item @code{algebraic}
3508 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3512 To determine whether an expression is commutative or non-commutative and if
3513 so, with which other expressions it would commutate, you use the methods
3514 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3515 for an explanation of these.
3518 @subsection Accessing subexpressions
3521 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3522 @code{function}, act as containers for subexpressions. For example, the
3523 subexpressions of a sum (an @code{add} object) are the individual terms,
3524 and the subexpressions of a @code{function} are the function's arguments.
3526 @cindex @code{nops()}
3528 GiNaC provides several ways of accessing subexpressions. The first way is to
3533 ex ex::op(size_t i);
3536 @code{nops()} determines the number of subexpressions (operands) contained
3537 in the expression, while @code{op(i)} returns the @code{i}-th
3538 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3539 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3540 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3541 @math{i>0} are the indices.
3544 @cindex @code{const_iterator}
3545 The second way to access subexpressions is via the STL-style random-access
3546 iterator class @code{const_iterator} and the methods
3549 const_iterator ex::begin();
3550 const_iterator ex::end();
3553 @code{begin()} returns an iterator referring to the first subexpression;
3554 @code{end()} returns an iterator which is one-past the last subexpression.
3555 If the expression has no subexpressions, then @code{begin() == end()}. These
3556 iterators can also be used in conjunction with non-modifying STL algorithms.
3558 Here is an example that (non-recursively) prints the subexpressions of a
3559 given expression in three different ways:
3566 for (size_t i = 0; i != e.nops(); ++i)
3567 cout << e.op(i) << endl;
3570 for (const_iterator i = e.begin(); i != e.end(); ++i)
3573 // with iterators and STL copy()
3574 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3578 @cindex @code{const_preorder_iterator}
3579 @cindex @code{const_postorder_iterator}
3580 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3581 expression's immediate children. GiNaC provides two additional iterator
3582 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3583 that iterate over all objects in an expression tree, in preorder or postorder,
3584 respectively. They are STL-style forward iterators, and are created with the
3588 const_preorder_iterator ex::preorder_begin();
3589 const_preorder_iterator ex::preorder_end();
3590 const_postorder_iterator ex::postorder_begin();
3591 const_postorder_iterator ex::postorder_end();
3594 The following example illustrates the differences between
3595 @code{const_iterator}, @code{const_preorder_iterator}, and
3596 @code{const_postorder_iterator}:
3600 symbol A("A"), B("B"), C("C");
3601 ex e = lst(lst(A, B), C);
3603 std::copy(e.begin(), e.end(),
3604 std::ostream_iterator<ex>(cout, "\n"));
3608 std::copy(e.preorder_begin(), e.preorder_end(),
3609 std::ostream_iterator<ex>(cout, "\n"));
3616 std::copy(e.postorder_begin(), e.postorder_end(),
3617 std::ostream_iterator<ex>(cout, "\n"));
3626 @cindex @code{relational} (class)
3627 Finally, the left-hand side and right-hand side expressions of objects of
3628 class @code{relational} (and only of these) can also be accessed with the
3637 @subsection Comparing expressions
3638 @cindex @code{is_equal()}
3639 @cindex @code{is_zero()}
3641 Expressions can be compared with the usual C++ relational operators like
3642 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3643 the result is usually not determinable and the result will be @code{false},
3644 except in the case of the @code{!=} operator. You should also be aware that
3645 GiNaC will only do the most trivial test for equality (subtracting both
3646 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3649 Actually, if you construct an expression like @code{a == b}, this will be
3650 represented by an object of the @code{relational} class (@pxref{Relations})
3651 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3653 There are also two methods
3656 bool ex::is_equal(const ex & other);
3660 for checking whether one expression is equal to another, or equal to zero,
3664 @subsection Ordering expressions
3665 @cindex @code{ex_is_less} (class)
3666 @cindex @code{ex_is_equal} (class)
3667 @cindex @code{compare()}
3669 Sometimes it is necessary to establish a mathematically well-defined ordering
3670 on a set of arbitrary expressions, for example to use expressions as keys
3671 in a @code{std::map<>} container, or to bring a vector of expressions into
3672 a canonical order (which is done internally by GiNaC for sums and products).
3674 The operators @code{<}, @code{>} etc. described in the last section cannot
3675 be used for this, as they don't implement an ordering relation in the
3676 mathematical sense. In particular, they are not guaranteed to be
3677 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3678 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3681 By default, STL classes and algorithms use the @code{<} and @code{==}
3682 operators to compare objects, which are unsuitable for expressions, but GiNaC
3683 provides two functors that can be supplied as proper binary comparison
3684 predicates to the STL:
3687 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3689 bool operator()(const ex &lh, const ex &rh) const;
3692 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3694 bool operator()(const ex &lh, const ex &rh) const;
3698 For example, to define a @code{map} that maps expressions to strings you
3702 std::map<ex, std::string, ex_is_less> myMap;
3705 Omitting the @code{ex_is_less} template parameter will introduce spurious
3706 bugs because the map operates improperly.
3708 Other examples for the use of the functors:
3716 std::sort(v.begin(), v.end(), ex_is_less());
3718 // count the number of expressions equal to '1'
3719 unsigned num_ones = std::count_if(v.begin(), v.end(),
3720 std::bind2nd(ex_is_equal(), 1));
3723 The implementation of @code{ex_is_less} uses the member function
3726 int ex::compare(const ex & other) const;
3729 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3730 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3734 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3735 @c node-name, next, previous, up
3736 @section Numerical Evaluation
3737 @cindex @code{evalf()}
3739 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3740 To evaluate them using floating-point arithmetic you need to call
3743 ex ex::evalf(int level = 0) const;
3746 @cindex @code{Digits}
3747 The accuracy of the evaluation is controlled by the global object @code{Digits}
3748 which can be assigned an integer value. The default value of @code{Digits}
3749 is 17. @xref{Numbers}, for more information and examples.
3751 To evaluate an expression to a @code{double} floating-point number you can
3752 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3756 // Approximate sin(x/Pi)
3758 ex e = series(sin(x/Pi), x == 0, 6);
3760 // Evaluate numerically at x=0.1
3761 ex f = evalf(e.subs(x == 0.1));
3763 // ex_to<numeric> is an unsafe cast, so check the type first
3764 if (is_a<numeric>(f)) @{
3765 double d = ex_to<numeric>(f).to_double();
3774 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3775 @c node-name, next, previous, up
3776 @section Substituting expressions
3777 @cindex @code{subs()}
3779 Algebraic objects inside expressions can be replaced with arbitrary
3780 expressions via the @code{.subs()} method:
3783 ex ex::subs(const ex & e, unsigned options = 0);
3784 ex ex::subs(const exmap & m, unsigned options = 0);
3785 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3788 In the first form, @code{subs()} accepts a relational of the form
3789 @samp{object == expression} or a @code{lst} of such relationals:
3793 symbol x("x"), y("y");
3795 ex e1 = 2*x^2-4*x+3;
3796 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3800 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3805 If you specify multiple substitutions, they are performed in parallel, so e.g.
3806 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3808 The second form of @code{subs()} takes an @code{exmap} object which is a
3809 pair associative container that maps expressions to expressions (currently
3810 implemented as a @code{std::map}). This is the most efficient one of the
3811 three @code{subs()} forms and should be used when the number of objects to
3812 be substituted is large or unknown.
3814 Using this form, the second example from above would look like this:
3818 symbol x("x"), y("y");
3824 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3828 The third form of @code{subs()} takes two lists, one for the objects to be
3829 replaced and one for the expressions to be substituted (both lists must
3830 contain the same number of elements). Using this form, you would write
3834 symbol x("x"), y("y");
3837 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3841 The optional last argument to @code{subs()} is a combination of
3842 @code{subs_options} flags. There are two options available:
3843 @code{subs_options::no_pattern} disables pattern matching, which makes
3844 large @code{subs()} operations significantly faster if you are not using
3845 patterns. The second option, @code{subs_options::algebraic} enables
3846 algebraic substitutions in products and powers.
3847 @ref{Pattern Matching and Advanced Substitutions}, for more information
3848 about patterns and algebraic substitutions.
3850 @code{subs()} performs syntactic substitution of any complete algebraic
3851 object; it does not try to match sub-expressions as is demonstrated by the
3856 symbol x("x"), y("y"), z("z");
3858 ex e1 = pow(x+y, 2);
3859 cout << e1.subs(x+y == 4) << endl;
3862 ex e2 = sin(x)*sin(y)*cos(x);
3863 cout << e2.subs(sin(x) == cos(x)) << endl;
3864 // -> cos(x)^2*sin(y)
3867 cout << e3.subs(x+y == 4) << endl;
3869 // (and not 4+z as one might expect)
3873 A more powerful form of substitution using wildcards is described in the
3877 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3878 @c node-name, next, previous, up
3879 @section Pattern matching and advanced substitutions
3880 @cindex @code{wildcard} (class)
3881 @cindex Pattern matching
3883 GiNaC allows the use of patterns for checking whether an expression is of a
3884 certain form or contains subexpressions of a certain form, and for
3885 substituting expressions in a more general way.
3887 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3888 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3889 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3890 an unsigned integer number to allow having multiple different wildcards in a
3891 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3892 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3896 ex wild(unsigned label = 0);
3899 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3902 Some examples for patterns:
3904 @multitable @columnfractions .5 .5
3905 @item @strong{Constructed as} @tab @strong{Output as}
3906 @item @code{wild()} @tab @samp{$0}
3907 @item @code{pow(x,wild())} @tab @samp{x^$0}
3908 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3909 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3915 @item Wildcards behave like symbols and are subject to the same algebraic
3916 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3917 @item As shown in the last example, to use wildcards for indices you have to
3918 use them as the value of an @code{idx} object. This is because indices must
3919 always be of class @code{idx} (or a subclass).
3920 @item Wildcards only represent expressions or subexpressions. It is not
3921 possible to use them as placeholders for other properties like index
3922 dimension or variance, representation labels, symmetry of indexed objects
3924 @item Because wildcards are commutative, it is not possible to use wildcards
3925 as part of noncommutative products.
3926 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3927 are also valid patterns.
3930 @subsection Matching expressions
3931 @cindex @code{match()}
3932 The most basic application of patterns is to check whether an expression
3933 matches a given pattern. This is done by the function
3936 bool ex::match(const ex & pattern);
3937 bool ex::match(const ex & pattern, lst & repls);
3940 This function returns @code{true} when the expression matches the pattern
3941 and @code{false} if it doesn't. If used in the second form, the actual
3942 subexpressions matched by the wildcards get returned in the @code{repls}
3943 object as a list of relations of the form @samp{wildcard == expression}.
3944 If @code{match()} returns false, the state of @code{repls} is undefined.
3945 For reproducible results, the list should be empty when passed to
3946 @code{match()}, but it is also possible to find similarities in multiple
3947 expressions by passing in the result of a previous match.
3949 The matching algorithm works as follows:
3952 @item A single wildcard matches any expression. If one wildcard appears
3953 multiple times in a pattern, it must match the same expression in all
3954 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3955 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3956 @item If the expression is not of the same class as the pattern, the match
3957 fails (i.e. a sum only matches a sum, a function only matches a function,
3959 @item If the pattern is a function, it only matches the same function
3960 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3961 @item Except for sums and products, the match fails if the number of
3962 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3964 @item If there are no subexpressions, the expressions and the pattern must
3965 be equal (in the sense of @code{is_equal()}).
3966 @item Except for sums and products, each subexpression (@code{op()}) must
3967 match the corresponding subexpression of the pattern.
3970 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3971 account for their commutativity and associativity:
3974 @item If the pattern contains a term or factor that is a single wildcard,
3975 this one is used as the @dfn{global wildcard}. If there is more than one
3976 such wildcard, one of them is chosen as the global wildcard in a random
3978 @item Every term/factor of the pattern, except the global wildcard, is
3979 matched against every term of the expression in sequence. If no match is
3980 found, the whole match fails. Terms that did match are not considered in
3982 @item If there are no unmatched terms left, the match succeeds. Otherwise
3983 the match fails unless there is a global wildcard in the pattern, in
3984 which case this wildcard matches the remaining terms.
3987 In general, having more than one single wildcard as a term of a sum or a
3988 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3991 Here are some examples in @command{ginsh} to demonstrate how it works (the
3992 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3993 match fails, and the list of wildcard replacements otherwise):
3996 > match((x+y)^a,(x+y)^a);
3998 > match((x+y)^a,(x+y)^b);
4000 > match((x+y)^a,$1^$2);
4002 > match((x+y)^a,$1^$1);
4004 > match((x+y)^(x+y),$1^$1);
4006 > match((x+y)^(x+y),$1^$2);
4008 > match((a+b)*(a+c),($1+b)*($1+c));
4010 > match((a+b)*(a+c),(a+$1)*(a+$2));
4012 (Unpredictable. The result might also be [$1==c,$2==b].)
4013 > match((a+b)*(a+c),($1+$2)*($1+$3));
4014 (The result is undefined. Due to the sequential nature of the algorithm
4015 and the re-ordering of terms in GiNaC, the match for the first factor
4016 may be @{$1==a,$2==b@} in which case the match for the second factor
4017 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4019 > match(a*(x+y)+a*z+b,a*$1+$2);
4020 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4021 @{$1=x+y,$2=a*z+b@}.)
4022 > match(a+b+c+d+e+f,c);
4024 > match(a+b+c+d+e+f,c+$0);
4026 > match(a+b+c+d+e+f,c+e+$0);
4028 > match(a+b,a+b+$0);
4030 > match(a*b^2,a^$1*b^$2);
4032 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4033 even though a==a^1.)
4034 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4036 > match(atan2(y,x^2),atan2(y,$0));
4040 @subsection Matching parts of expressions
4041 @cindex @code{has()}
4042 A more general way to look for patterns in expressions is provided by the
4046 bool ex::has(const ex & pattern);
4049 This function checks whether a pattern is matched by an expression itself or
4050 by any of its subexpressions.
4052 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4053 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4056 > has(x*sin(x+y+2*a),y);
4058 > has(x*sin(x+y+2*a),x+y);
4060 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4061 has the subexpressions "x", "y" and "2*a".)
4062 > has(x*sin(x+y+2*a),x+y+$1);
4064 (But this is possible.)
4065 > has(x*sin(2*(x+y)+2*a),x+y);
4067 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4068 which "x+y" is not a subexpression.)
4071 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4073 > has(4*x^2-x+3,$1*x);
4075 > has(4*x^2+x+3,$1*x);
4077 (Another possible pitfall. The first expression matches because the term
4078 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4079 contains a linear term you should use the coeff() function instead.)
4082 @cindex @code{find()}
4086 bool ex::find(const ex & pattern, lst & found);
4089 works a bit like @code{has()} but it doesn't stop upon finding the first
4090 match. Instead, it appends all found matches to the specified list. If there
4091 are multiple occurrences of the same expression, it is entered only once to
4092 the list. @code{find()} returns false if no matches were found (in
4093 @command{ginsh}, it returns an empty list):
4096 > find(1+x+x^2+x^3,x);
4098 > find(1+x+x^2+x^3,y);
4100 > find(1+x+x^2+x^3,x^$1);
4102 (Note the absence of "x".)
4103 > expand((sin(x)+sin(y))*(a+b));
4104 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4109 @subsection Substituting expressions
4110 @cindex @code{subs()}
4111 Probably the most useful application of patterns is to use them for
4112 substituting expressions with the @code{subs()} method. Wildcards can be
4113 used in the search patterns as well as in the replacement expressions, where
4114 they get replaced by the expressions matched by them. @code{subs()} doesn't
4115 know anything about algebra; it performs purely syntactic substitutions.
4120 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4122 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4124 > subs((a+b+c)^2,a+b==x);
4126 > subs((a+b+c)^2,a+b+$1==x+$1);
4128 > subs(a+2*b,a+b==x);
4130 > subs(4*x^3-2*x^2+5*x-1,x==a);
4132 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4134 > subs(sin(1+sin(x)),sin($1)==cos($1));
4136 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4140 The last example would be written in C++ in this way:
4144 symbol a("a"), b("b"), x("x"), y("y");
4145 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4146 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4147 cout << e.expand() << endl;
4152 @subsection Algebraic substitutions
4153 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4154 enables smarter, algebraic substitutions in products and powers. If you want
4155 to substitute some factors of a product, you only need to list these factors
4156 in your pattern. Furthermore, if an (integer) power of some expression occurs
4157 in your pattern and in the expression that you want the substitution to occur
4158 in, it can be substituted as many times as possible, without getting negative
4161 An example clarifies it all (hopefully):
4164 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4165 subs_options::algebraic) << endl;
4166 // --> (y+x)^6+b^6+a^6
4168 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4170 // Powers and products are smart, but addition is just the same.
4172 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4175 // As I said: addition is just the same.
4177 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4178 // --> x^3*b*a^2+2*b
4180 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4182 // --> 2*b+x^3*b^(-1)*a^(-2)
4184 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4185 // --> -1-2*a^2+4*a^3+5*a
4187 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4188 subs_options::algebraic) << endl;
4189 // --> -1+5*x+4*x^3-2*x^2
4190 // You should not really need this kind of patterns very often now.
4191 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4193 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4194 subs_options::algebraic) << endl;
4195 // --> cos(1+cos(x))
4197 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4198 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4199 subs_options::algebraic)) << endl;
4204 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4205 @c node-name, next, previous, up
4206 @section Applying a Function on Subexpressions
4207 @cindex tree traversal
4208 @cindex @code{map()}
4210 Sometimes you may want to perform an operation on specific parts of an
4211 expression while leaving the general structure of it intact. An example
4212 of this would be a matrix trace operation: the trace of a sum is the sum
4213 of the traces of the individual terms. That is, the trace should @dfn{map}
4214 on the sum, by applying itself to each of the sum's operands. It is possible
4215 to do this manually which usually results in code like this:
4220 if (is_a<matrix>(e))
4221 return ex_to<matrix>(e).trace();
4222 else if (is_a<add>(e)) @{
4224 for (size_t i=0; i<e.nops(); i++)
4225 sum += calc_trace(e.op(i));
4227 @} else if (is_a<mul>)(e)) @{
4235 This is, however, slightly inefficient (if the sum is very large it can take
4236 a long time to add the terms one-by-one), and its applicability is limited to
4237 a rather small class of expressions. If @code{calc_trace()} is called with
4238 a relation or a list as its argument, you will probably want the trace to
4239 be taken on both sides of the relation or of all elements of the list.
4241 GiNaC offers the @code{map()} method to aid in the implementation of such
4245 ex ex::map(map_function & f) const;
4246 ex ex::map(ex (*f)(const ex & e)) const;
4249 In the first (preferred) form, @code{map()} takes a function object that
4250 is subclassed from the @code{map_function} class. In the second form, it
4251 takes a pointer to a function that accepts and returns an expression.
4252 @code{map()} constructs a new expression of the same type, applying the
4253 specified function on all subexpressions (in the sense of @code{op()}),
4256 The use of a function object makes it possible to supply more arguments to
4257 the function that is being mapped, or to keep local state information.
4258 The @code{map_function} class declares a virtual function call operator
4259 that you can overload. Here is a sample implementation of @code{calc_trace()}
4260 that uses @code{map()} in a recursive fashion:
4263 struct calc_trace : public map_function @{
4264 ex operator()(const ex &e)
4266 if (is_a<matrix>(e))
4267 return ex_to<matrix>(e).trace();
4268 else if (is_a<mul>(e)) @{
4271 return e.map(*this);
4276 This function object could then be used like this:
4280 ex M = ... // expression with matrices
4281 calc_trace do_trace;
4282 ex tr = do_trace(M);
4286 Here is another example for you to meditate over. It removes quadratic
4287 terms in a variable from an expanded polynomial:
4290 struct map_rem_quad : public map_function @{
4292 map_rem_quad(const ex & var_) : var(var_) @{@}
4294 ex operator()(const ex & e)
4296 if (is_a<add>(e) || is_a<mul>(e))
4297 return e.map(*this);
4298 else if (is_a<power>(e) &&
4299 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4309 symbol x("x"), y("y");
4312 for (int i=0; i<8; i++)
4313 e += pow(x, i) * pow(y, 8-i) * (i+1);
4315 // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8
4317 map_rem_quad rem_quad(x);
4318 cout << rem_quad(e) << endl;
4319 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4323 @command{ginsh} offers a slightly different implementation of @code{map()}
4324 that allows applying algebraic functions to operands. The second argument
4325 to @code{map()} is an expression containing the wildcard @samp{$0} which
4326 acts as the placeholder for the operands:
4331 > map(a+2*b,sin($0));
4333 > map(@{a,b,c@},$0^2+$0);
4334 @{a^2+a,b^2+b,c^2+c@}
4337 Note that it is only possible to use algebraic functions in the second
4338 argument. You can not use functions like @samp{diff()}, @samp{op()},
4339 @samp{subs()} etc. because these are evaluated immediately:
4342 > map(@{a,b,c@},diff($0,a));
4344 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4345 to "map(@{a,b,c@},0)".
4349 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4350 @c node-name, next, previous, up
4351 @section Visitors and Tree Traversal
4352 @cindex tree traversal
4353 @cindex @code{visitor} (class)
4354 @cindex @code{accept()}
4355 @cindex @code{visit()}
4356 @cindex @code{traverse()}
4357 @cindex @code{traverse_preorder()}
4358 @cindex @code{traverse_postorder()}
4360 Suppose that you need a function that returns a list of all indices appearing
4361 in an arbitrary expression. The indices can have any dimension, and for
4362 indices with variance you always want the covariant version returned.
4364 You can't use @code{get_free_indices()} because you also want to include
4365 dummy indices in the list, and you can't use @code{find()} as it needs
4366 specific index dimensions (and it would require two passes: one for indices
4367 with variance, one for plain ones).
4369 The obvious solution to this problem is a tree traversal with a type switch,
4370 such as the following:
4373 void gather_indices_helper(const ex & e, lst & l)
4375 if (is_a<varidx>(e)) @{
4376 const varidx & vi = ex_to<varidx>(e);
4377 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4378 @} else if (is_a<idx>(e)) @{
4381 size_t n = e.nops();
4382 for (size_t i = 0; i < n; ++i)
4383 gather_indices_helper(e.op(i), l);
4387 lst gather_indices(const ex & e)
4390 gather_indices_helper(e, l);
4397 This works fine but fans of object-oriented programming will feel
4398 uncomfortable with the type switch. One reason is that there is a possibility
4399 for subtle bugs regarding derived classes. If we had, for example, written
4402 if (is_a<idx>(e)) @{
4404 @} else if (is_a<varidx>(e)) @{
4408 in @code{gather_indices_helper}, the code wouldn't have worked because the
4409 first line "absorbs" all classes derived from @code{idx}, including
4410 @code{varidx}, so the special case for @code{varidx} would never have been
4413 Also, for a large number of classes, a type switch like the above can get
4414 unwieldy and inefficient (it's a linear search, after all).
4415 @code{gather_indices_helper} only checks for two classes, but if you had to
4416 write a function that required a different implementation for nearly
4417 every GiNaC class, the result would be very hard to maintain and extend.
4419 The cleanest approach to the problem would be to add a new virtual function
4420 to GiNaC's class hierarchy. In our example, there would be specializations
4421 for @code{idx} and @code{varidx} while the default implementation in
4422 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4423 impossible to add virtual member functions to existing classes without
4424 changing their source and recompiling everything. GiNaC comes with source,
4425 so you could actually do this, but for a small algorithm like the one
4426 presented this would be impractical.
4428 One solution to this dilemma is the @dfn{Visitor} design pattern,
4429 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4430 variation, described in detail in
4431 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4432 virtual functions to the class hierarchy to implement operations, GiNaC
4433 provides a single "bouncing" method @code{accept()} that takes an instance
4434 of a special @code{visitor} class and redirects execution to the one
4435 @code{visit()} virtual function of the visitor that matches the type of
4436 object that @code{accept()} was being invoked on.
4438 Visitors in GiNaC must derive from the global @code{visitor} class as well
4439 as from the class @code{T::visitor} of each class @code{T} they want to
4440 visit, and implement the member functions @code{void visit(const T &)} for
4446 void ex::accept(visitor & v) const;
4449 will then dispatch to the correct @code{visit()} member function of the
4450 specified visitor @code{v} for the type of GiNaC object at the root of the
4451 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4453 Here is an example of a visitor:
4457 : public visitor, // this is required
4458 public add::visitor, // visit add objects
4459 public numeric::visitor, // visit numeric objects
4460 public basic::visitor // visit basic objects
4462 void visit(const add & x)
4463 @{ cout << "called with an add object" << endl; @}
4465 void visit(const numeric & x)
4466 @{ cout << "called with a numeric object" << endl; @}
4468 void visit(const basic & x)
4469 @{ cout << "called with a basic object" << endl; @}
4473 which can be used as follows:
4484 // prints "called with a numeric object"
4486 // prints "called with an add object"
4488 // prints "called with a basic object"
4492 The @code{visit(const basic &)} method gets called for all objects that are
4493 not @code{numeric} or @code{add} and acts as an (optional) default.
4495 From a conceptual point of view, the @code{visit()} methods of the visitor
4496 behave like a newly added virtual function of the visited hierarchy.
4497 In addition, visitors can store state in member variables, and they can
4498 be extended by deriving a new visitor from an existing one, thus building
4499 hierarchies of visitors.
4501 We can now rewrite our index example from above with a visitor:
4504 class gather_indices_visitor
4505 : public visitor, public idx::visitor, public varidx::visitor
4509 void visit(const idx & i)
4514 void visit(const varidx & vi)
4516 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4520 const lst & get_result() // utility function
4529 What's missing is the tree traversal. We could implement it in
4530 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4533 void ex::traverse_preorder(visitor & v) const;
4534 void ex::traverse_postorder(visitor & v) const;
4535 void ex::traverse(visitor & v) const;
4538 @code{traverse_preorder()} visits a node @emph{before} visiting its
4539 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4540 visiting its subexpressions. @code{traverse()} is a synonym for
4541 @code{traverse_preorder()}.
4543 Here is a new implementation of @code{gather_indices()} that uses the visitor
4544 and @code{traverse()}:
4547 lst gather_indices(const ex & e)
4549 gather_indices_visitor v;
4551 return v.get_result();
4555 Alternatively, you could use pre- or postorder iterators for the tree
4559 lst gather_indices(const ex & e)
4561 gather_indices_visitor v;
4562 for (const_preorder_iterator i = e.preorder_begin();
4563 i != e.preorder_end(); ++i) @{
4566 return v.get_result();
4571 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4572 @c node-name, next, previous, up
4573 @section Polynomial arithmetic
4575 @subsection Expanding and collecting
4576 @cindex @code{expand()}
4577 @cindex @code{collect()}
4578 @cindex @code{collect_common_factors()}
4580 A polynomial in one or more variables has many equivalent
4581 representations. Some useful ones serve a specific purpose. Consider
4582 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4583 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4584 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4585 representations are the recursive ones where one collects for exponents
4586 in one of the three variable. Since the factors are themselves
4587 polynomials in the remaining two variables the procedure can be
4588 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4589 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4592 To bring an expression into expanded form, its method
4595 ex ex::expand(unsigned options = 0);
4598 may be called. In our example above, this corresponds to @math{4*x*y +
4599 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4600 GiNaC is not easy to guess you should be prepared to see different
4601 orderings of terms in such sums!
4603 Another useful representation of multivariate polynomials is as a
4604 univariate polynomial in one of the variables with the coefficients
4605 being polynomials in the remaining variables. The method
4606 @code{collect()} accomplishes this task:
4609 ex ex::collect(const ex & s, bool distributed = false);
4612 The first argument to @code{collect()} can also be a list of objects in which
4613 case the result is either a recursively collected polynomial, or a polynomial
4614 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4615 by the @code{distributed} flag.
4617 Note that the original polynomial needs to be in expanded form (for the
4618 variables concerned) in order for @code{collect()} to be able to find the
4619 coefficients properly.
4621 The following @command{ginsh} transcript shows an application of @code{collect()}
4622 together with @code{find()}:
4625 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4626 d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)+q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4627 > collect(a,@{p,q@});
4628 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4629 > collect(a,find(a,sin($1)));
4630 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4631 > collect(a,@{find(a,sin($1)),p,q@});
4632 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4633 > collect(a,@{find(a,sin($1)),d@});
4634 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4637 Polynomials can often be brought into a more compact form by collecting
4638 common factors from the terms of sums. This is accomplished by the function
4641 ex collect_common_factors(const ex & e);
4644 This function doesn't perform a full factorization but only looks for
4645 factors which are already explicitly present:
4648 > collect_common_factors(a*x+a*y);
4650 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4652 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4653 (c+a)*a*(x*y+y^2+x)*b
4656 @subsection Degree and coefficients
4657 @cindex @code{degree()}
4658 @cindex @code{ldegree()}
4659 @cindex @code{coeff()}
4661 The degree and low degree of a polynomial can be obtained using the two
4665 int ex::degree(const ex & s);
4666 int ex::ldegree(const ex & s);
4669 which also work reliably on non-expanded input polynomials (they even work
4670 on rational functions, returning the asymptotic degree). By definition, the
4671 degree of zero is zero. To extract a coefficient with a certain power from
4672 an expanded polynomial you use
4675 ex ex::coeff(const ex & s, int n);
4678 You can also obtain the leading and trailing coefficients with the methods
4681 ex ex::lcoeff(const ex & s);
4682 ex ex::tcoeff(const ex & s);
4685 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4688 An application is illustrated in the next example, where a multivariate
4689 polynomial is analyzed:
4693 symbol x("x"), y("y");
4694 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4695 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4696 ex Poly = PolyInp.expand();
4698 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4699 cout << "The x^" << i << "-coefficient is "
4700 << Poly.coeff(x,i) << endl;
4702 cout << "As polynomial in y: "
4703 << Poly.collect(y) << endl;
4707 When run, it returns an output in the following fashion:
4710 The x^0-coefficient is y^2+11*y
4711 The x^1-coefficient is 5*y^2-2*y
4712 The x^2-coefficient is -1
4713 The x^3-coefficient is 4*y
4714 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4717 As always, the exact output may vary between different versions of GiNaC
4718 or even from run to run since the internal canonical ordering is not
4719 within the user's sphere of influence.
4721 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4722 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4723 with non-polynomial expressions as they not only work with symbols but with
4724 constants, functions and indexed objects as well:
4728 symbol a("a"), b("b"), c("c"), x("x");
4729 idx i(symbol("i"), 3);
4731 ex e = pow(sin(x) - cos(x), 4);
4732 cout << e.degree(cos(x)) << endl;
4734 cout << e.expand().coeff(sin(x), 3) << endl;
4737 e = indexed(a+b, i) * indexed(b+c, i);
4738 e = e.expand(expand_options::expand_indexed);
4739 cout << e.collect(indexed(b, i)) << endl;
4740 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4745 @subsection Polynomial division
4746 @cindex polynomial division
4749 @cindex pseudo-remainder
4750 @cindex @code{quo()}
4751 @cindex @code{rem()}
4752 @cindex @code{prem()}
4753 @cindex @code{divide()}
4758 ex quo(const ex & a, const ex & b, const ex & x);
4759 ex rem(const ex & a, const ex & b, const ex & x);
4762 compute the quotient and remainder of univariate polynomials in the variable
4763 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4765 The additional function
4768 ex prem(const ex & a, const ex & b, const ex & x);
4771 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4772 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4774 Exact division of multivariate polynomials is performed by the function
4777 bool divide(const ex & a, const ex & b, ex & q);
4780 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4781 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4782 in which case the value of @code{q} is undefined.
4785 @subsection Unit, content and primitive part
4786 @cindex @code{unit()}
4787 @cindex @code{content()}
4788 @cindex @code{primpart()}
4789 @cindex @code{unitcontprim()}
4794 ex ex::unit(const ex & x);
4795 ex ex::content(const ex & x);
4796 ex ex::primpart(const ex & x);
4797 ex ex::primpart(const ex & x, const ex & c);
4800 return the unit part, content part, and primitive polynomial of a multivariate
4801 polynomial with respect to the variable @samp{x} (the unit part being the sign
4802 of the leading coefficient, the content part being the GCD of the coefficients,
4803 and the primitive polynomial being the input polynomial divided by the unit and
4804 content parts). The second variant of @code{primpart()} expects the previously
4805 calculated content part of the polynomial in @code{c}, which enables it to
4806 work faster in the case where the content part has already been computed. The
4807 product of unit, content, and primitive part is the original polynomial.
4809 Additionally, the method
4812 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
4815 computes the unit, content, and primitive parts in one go, returning them
4816 in @code{u}, @code{c}, and @code{p}, respectively.
4819 @subsection GCD, LCM and resultant
4822 @cindex @code{gcd()}
4823 @cindex @code{lcm()}
4825 The functions for polynomial greatest common divisor and least common
4826 multiple have the synopsis
4829 ex gcd(const ex & a, const ex & b);
4830 ex lcm(const ex & a, const ex & b);
4833 The functions @code{gcd()} and @code{lcm()} accept two expressions
4834 @code{a} and @code{b} as arguments and return a new expression, their
4835 greatest common divisor or least common multiple, respectively. If the
4836 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4837 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4840 #include <ginac/ginac.h>
4841 using namespace GiNaC;
4845 symbol x("x"), y("y"), z("z");
4846 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4847 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4849 ex P_gcd = gcd(P_a, P_b);
4851 ex P_lcm = lcm(P_a, P_b);
4852 // 4*x*y^2 + 13*y*x*z + 20*y^3 + 81*y^2*z + 67*y*z^2 + 3*x*z^2 + 12*z^3
4857 @cindex @code{resultant()}
4859 The resultant of two expressions only makes sense with polynomials.
4860 It is always computed with respect to a specific symbol within the
4861 expressions. The function has the interface
4864 ex resultant(const ex & a, const ex & b, const ex & s);
4867 Resultants are symmetric in @code{a} and @code{b}. The following example
4868 computes the resultant of two expressions with respect to @code{x} and
4869 @code{y}, respectively:
4872 #include <ginac/ginac.h>
4873 using namespace GiNaC;
4877 symbol x("x"), y("y");
4879 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
4882 r = resultant(e1, e2, x);
4884 r = resultant(e1, e2, y);
4889 @subsection Square-free decomposition
4890 @cindex square-free decomposition
4891 @cindex factorization
4892 @cindex @code{sqrfree()}
4894 GiNaC still lacks proper factorization support. Some form of
4895 factorization is, however, easily implemented by noting that factors
4896 appearing in a polynomial with power two or more also appear in the
4897 derivative and hence can easily be found by computing the GCD of the
4898 original polynomial and its derivatives. Any decent system has an
4899 interface for this so called square-free factorization. So we provide
4902 ex sqrfree(const ex & a, const lst & l = lst());
4904 Here is an example that by the way illustrates how the exact form of the
4905 result may slightly depend on the order of differentiation, calling for
4906 some care with subsequent processing of the result:
4909 symbol x("x"), y("y");
4910 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4912 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4913 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4915 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4916 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4918 cout << sqrfree(BiVarPol) << endl;
4919 // -> depending on luck, any of the above
4922 Note also, how factors with the same exponents are not fully factorized
4926 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4927 @c node-name, next, previous, up
4928 @section Rational expressions
4930 @subsection The @code{normal} method
4931 @cindex @code{normal()}
4932 @cindex simplification
4933 @cindex temporary replacement
4935 Some basic form of simplification of expressions is called for frequently.
4936 GiNaC provides the method @code{.normal()}, which converts a rational function
4937 into an equivalent rational function of the form @samp{numerator/denominator}
4938 where numerator and denominator are coprime. If the input expression is already
4939 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4940 otherwise it performs fraction addition and multiplication.
4942 @code{.normal()} can also be used on expressions which are not rational functions
4943 as it will replace all non-rational objects (like functions or non-integer
4944 powers) by temporary symbols to bring the expression to the domain of rational
4945 functions before performing the normalization, and re-substituting these
4946 symbols afterwards. This algorithm is also available as a separate method
4947 @code{.to_rational()}, described below.
4949 This means that both expressions @code{t1} and @code{t2} are indeed
4950 simplified in this little code snippet:
4955 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4956 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4957 std::cout << "t1 is " << t1.normal() << std::endl;
4958 std::cout << "t2 is " << t2.normal() << std::endl;
4962 Of course this works for multivariate polynomials too, so the ratio of
4963 the sample-polynomials from the section about GCD and LCM above would be
4964 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4967 @subsection Numerator and denominator
4970 @cindex @code{numer()}
4971 @cindex @code{denom()}
4972 @cindex @code{numer_denom()}
4974 The numerator and denominator of an expression can be obtained with
4979 ex ex::numer_denom();
4982 These functions will first normalize the expression as described above and
4983 then return the numerator, denominator, or both as a list, respectively.
4984 If you need both numerator and denominator, calling @code{numer_denom()} is
4985 faster than using @code{numer()} and @code{denom()} separately.
4988 @subsection Converting to a polynomial or rational expression
4989 @cindex @code{to_polynomial()}
4990 @cindex @code{to_rational()}
4992 Some of the methods described so far only work on polynomials or rational
4993 functions. GiNaC provides a way to extend the domain of these functions to
4994 general expressions by using the temporary replacement algorithm described
4995 above. You do this by calling
4998 ex ex::to_polynomial(exmap & m);
4999 ex ex::to_polynomial(lst & l);
5003 ex ex::to_rational(exmap & m);
5004 ex ex::to_rational(lst & l);
5007 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5008 will be filled with the generated temporary symbols and their replacement
5009 expressions in a format that can be used directly for the @code{subs()}
5010 method. It can also already contain a list of replacements from an earlier
5011 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5012 possible to use it on multiple expressions and get consistent results.
5014 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5015 is probably best illustrated with an example:
5019 symbol x("x"), y("y");
5020 ex a = 2*x/sin(x) - y/(3*sin(x));
5024 ex p = a.to_polynomial(lp);
5025 cout << " = " << p << "\n with " << lp << endl;
5026 // = symbol3*symbol2*y+2*symbol2*x
5027 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5030 ex r = a.to_rational(lr);
5031 cout << " = " << r << "\n with " << lr << endl;
5032 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5033 // with @{symbol4==sin(x)@}
5037 The following more useful example will print @samp{sin(x)-cos(x)}:
5042 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5043 ex b = sin(x) + cos(x);
5046 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5047 cout << q.subs(m) << endl;
5052 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5053 @c node-name, next, previous, up
5054 @section Symbolic differentiation
5055 @cindex differentiation
5056 @cindex @code{diff()}
5058 @cindex product rule
5060 GiNaC's objects know how to differentiate themselves. Thus, a
5061 polynomial (class @code{add}) knows that its derivative is the sum of
5062 the derivatives of all the monomials:
5066 symbol x("x"), y("y"), z("z");
5067 ex P = pow(x, 5) + pow(x, 2) + y;
5069 cout << P.diff(x,2) << endl;
5071 cout << P.diff(y) << endl; // 1
5073 cout << P.diff(z) << endl; // 0
5078 If a second integer parameter @var{n} is given, the @code{diff} method
5079 returns the @var{n}th derivative.
5081 If @emph{every} object and every function is told what its derivative
5082 is, all derivatives of composed objects can be calculated using the
5083 chain rule and the product rule. Consider, for instance the expression
5084 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5085 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5086 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5087 out that the composition is the generating function for Euler Numbers,
5088 i.e. the so called @var{n}th Euler number is the coefficient of
5089 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5090 identity to code a function that generates Euler numbers in just three
5093 @cindex Euler numbers
5095 #include <ginac/ginac.h>
5096 using namespace GiNaC;
5098 ex EulerNumber(unsigned n)
5101 const ex generator = pow(cosh(x),-1);
5102 return generator.diff(x,n).subs(x==0);
5107 for (unsigned i=0; i<11; i+=2)
5108 std::cout << EulerNumber(i) << std::endl;
5113 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5114 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5115 @code{i} by two since all odd Euler numbers vanish anyways.
5118 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5119 @c node-name, next, previous, up
5120 @section Series expansion
5121 @cindex @code{series()}
5122 @cindex Taylor expansion
5123 @cindex Laurent expansion
5124 @cindex @code{pseries} (class)
5125 @cindex @code{Order()}
5127 Expressions know how to expand themselves as a Taylor series or (more
5128 generally) a Laurent series. As in most conventional Computer Algebra
5129 Systems, no distinction is made between those two. There is a class of
5130 its own for storing such series (@code{class pseries}) and a built-in
5131 function (called @code{Order}) for storing the order term of the series.
5132 As a consequence, if you want to work with series, i.e. multiply two
5133 series, you need to call the method @code{ex::series} again to convert
5134 it to a series object with the usual structure (expansion plus order
5135 term). A sample application from special relativity could read:
5138 #include <ginac/ginac.h>
5139 using namespace std;
5140 using namespace GiNaC;
5144 symbol v("v"), c("c");
5146 ex gamma = 1/sqrt(1 - pow(v/c,2));
5147 ex mass_nonrel = gamma.series(v==0, 10);
5149 cout << "the relativistic mass increase with v is " << endl
5150 << mass_nonrel << endl;
5152 cout << "the inverse square of this series is " << endl
5153 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5157 Only calling the series method makes the last output simplify to
5158 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5159 series raised to the power @math{-2}.
5161 @cindex Machin's formula
5162 As another instructive application, let us calculate the numerical
5163 value of Archimedes' constant
5167 (for which there already exists the built-in constant @code{Pi})
5168 using John Machin's amazing formula
5170 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5173 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5175 This equation (and similar ones) were used for over 200 years for
5176 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5177 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5178 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5179 order term with it and the question arises what the system is supposed
5180 to do when the fractions are plugged into that order term. The solution
5181 is to use the function @code{series_to_poly()} to simply strip the order
5185 #include <ginac/ginac.h>
5186 using namespace GiNaC;
5188 ex machin_pi(int degr)
5191 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5192 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5193 -4*pi_expansion.subs(x==numeric(1,239));
5199 using std::cout; // just for fun, another way of...
5200 using std::endl; // ...dealing with this namespace std.
5202 for (int i=2; i<12; i+=2) @{
5203 pi_frac = machin_pi(i);
5204 cout << i << ":\t" << pi_frac << endl
5205 << "\t" << pi_frac.evalf() << endl;
5211 Note how we just called @code{.series(x,degr)} instead of
5212 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5213 method @code{series()}: if the first argument is a symbol the expression
5214 is expanded in that symbol around point @code{0}. When you run this
5215 program, it will type out:
5219 3.1832635983263598326
5220 4: 5359397032/1706489875
5221 3.1405970293260603143
5222 6: 38279241713339684/12184551018734375
5223 3.141621029325034425
5224 8: 76528487109180192540976/24359780855939418203125
5225 3.141591772182177295
5226 10: 327853873402258685803048818236/104359128170408663038552734375
5227 3.1415926824043995174
5231 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5232 @c node-name, next, previous, up
5233 @section Symmetrization
5234 @cindex @code{symmetrize()}
5235 @cindex @code{antisymmetrize()}
5236 @cindex @code{symmetrize_cyclic()}
5241 ex ex::symmetrize(const lst & l);
5242 ex ex::antisymmetrize(const lst & l);
5243 ex ex::symmetrize_cyclic(const lst & l);
5246 symmetrize an expression by returning the sum over all symmetric,
5247 antisymmetric or cyclic permutations of the specified list of objects,
5248 weighted by the number of permutations.
5250 The three additional methods
5253 ex ex::symmetrize();
5254 ex ex::antisymmetrize();
5255 ex ex::symmetrize_cyclic();
5258 symmetrize or antisymmetrize an expression over its free indices.
5260 Symmetrization is most useful with indexed expressions but can be used with
5261 almost any kind of object (anything that is @code{subs()}able):
5265 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5266 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5268 cout << indexed(A, i, j).symmetrize() << endl;
5269 // -> 1/2*A.j.i+1/2*A.i.j
5270 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5271 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5272 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5273 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5277 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5278 @c node-name, next, previous, up
5279 @section Predefined mathematical functions
5281 @subsection Overview
5283 GiNaC contains the following predefined mathematical functions:
5286 @multitable @columnfractions .30 .70
5287 @item @strong{Name} @tab @strong{Function}
5290 @cindex @code{abs()}
5291 @item @code{csgn(x)}
5293 @cindex @code{conjugate()}
5294 @item @code{conjugate(x)}
5295 @tab complex conjugation
5296 @cindex @code{csgn()}
5297 @item @code{sqrt(x)}
5298 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5299 @cindex @code{sqrt()}
5302 @cindex @code{sin()}
5305 @cindex @code{cos()}
5308 @cindex @code{tan()}
5309 @item @code{asin(x)}
5311 @cindex @code{asin()}
5312 @item @code{acos(x)}
5314 @cindex @code{acos()}
5315 @item @code{atan(x)}
5316 @tab inverse tangent
5317 @cindex @code{atan()}
5318 @item @code{atan2(y, x)}
5319 @tab inverse tangent with two arguments
5320 @item @code{sinh(x)}
5321 @tab hyperbolic sine
5322 @cindex @code{sinh()}
5323 @item @code{cosh(x)}
5324 @tab hyperbolic cosine
5325 @cindex @code{cosh()}
5326 @item @code{tanh(x)}
5327 @tab hyperbolic tangent
5328 @cindex @code{tanh()}
5329 @item @code{asinh(x)}
5330 @tab inverse hyperbolic sine
5331 @cindex @code{asinh()}
5332 @item @code{acosh(x)}
5333 @tab inverse hyperbolic cosine
5334 @cindex @code{acosh()}
5335 @item @code{atanh(x)}
5336 @tab inverse hyperbolic tangent
5337 @cindex @code{atanh()}
5339 @tab exponential function
5340 @cindex @code{exp()}
5342 @tab natural logarithm
5343 @cindex @code{log()}
5346 @cindex @code{Li2()}
5347 @item @code{Li(m, x)}
5348 @tab classical polylogarithm as well as multiple polylogarithm
5350 @item @code{S(n, p, x)}
5351 @tab Nielsen's generalized polylogarithm
5353 @item @code{H(m, x)}
5354 @tab harmonic polylogarithm
5356 @item @code{zeta(m)}
5357 @tab Riemann's zeta function as well as multiple zeta value
5358 @cindex @code{zeta()}
5359 @item @code{zeta(m, s)}
5360 @tab alternating Euler sum
5361 @cindex @code{zeta()}
5362 @item @code{zetaderiv(n, x)}
5363 @tab derivatives of Riemann's zeta function
5364 @item @code{tgamma(x)}
5366 @cindex @code{tgamma()}
5367 @cindex gamma function
5368 @item @code{lgamma(x)}
5369 @tab logarithm of gamma function
5370 @cindex @code{lgamma()}
5371 @item @code{beta(x, y)}
5372 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5373 @cindex @code{beta()}
5375 @tab psi (digamma) function
5376 @cindex @code{psi()}
5377 @item @code{psi(n, x)}
5378 @tab derivatives of psi function (polygamma functions)
5379 @item @code{factorial(n)}
5380 @tab factorial function @math{n!}
5381 @cindex @code{factorial()}
5382 @item @code{binomial(n, k)}
5383 @tab binomial coefficients
5384 @cindex @code{binomial()}
5385 @item @code{Order(x)}
5386 @tab order term function in truncated power series
5387 @cindex @code{Order()}
5392 For functions that have a branch cut in the complex plane GiNaC follows
5393 the conventions for C++ as defined in the ANSI standard as far as
5394 possible. In particular: the natural logarithm (@code{log}) and the
5395 square root (@code{sqrt}) both have their branch cuts running along the
5396 negative real axis where the points on the axis itself belong to the
5397 upper part (i.e. continuous with quadrant II). The inverse
5398 trigonometric and hyperbolic functions are not defined for complex
5399 arguments by the C++ standard, however. In GiNaC we follow the
5400 conventions used by CLN, which in turn follow the carefully designed
5401 definitions in the Common Lisp standard. It should be noted that this
5402 convention is identical to the one used by the C99 standard and by most
5403 serious CAS. It is to be expected that future revisions of the C++
5404 standard incorporate these functions in the complex domain in a manner
5405 compatible with C99.
5407 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5408 @c node-name, next, previous, up
5409 @subsection Multiple polylogarithms
5411 @cindex polylogarithm
5412 @cindex Nielsen's generalized polylogarithm
5413 @cindex harmonic polylogarithm
5414 @cindex multiple zeta value
5415 @cindex alternating Euler sum
5416 @cindex multiple polylogarithm
5418 The multiple polylogarithm is the most generic member of a family of functions,
5419 to which others like the harmonic polylogarithm, Nielsen's generalized
5420 polylogarithm and the multiple zeta value belong.
5421 Everyone of these functions can also be written as a multiple polylogarithm with specific
5422 parameters. This whole family of functions is therefore often referred to simply as
5423 multiple polylogarithms, containing @code{Li}, @code{H}, @code{S} and @code{zeta}.
5425 To facilitate the discussion of these functions we distinguish between indices and
5426 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5427 @code{n} or @code{p}, whereas arguments are printed as @code{x}.
5429 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5430 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5431 for the argument @code{x} as well.
5432 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5433 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5434 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5435 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5436 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5438 The functions print in LaTeX format as
5440 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5446 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5449 $\zeta(m_1,m_2,\ldots,m_k)$.
5451 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5452 are printed with a line above, e.g.
5454 $\zeta(5,\overline{2})$.
5456 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5458 Definitions and analytical as well as numerical properties of multiple polylogarithms
5459 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5460 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5461 except for a few differences which will be explicitly stated in the following.
5463 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5464 that the indices and arguments are understood to be in the same order as in which they appear in
5465 the series representation. This means
5467 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5470 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5473 $\zeta(1,2)$ evaluates to infinity.
5475 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5478 The functions only evaluate if the indices are integers greater than zero, except for the indices
5479 @code{s} in @code{zeta} and @code{m} in @code{H}. Since @code{s} will be interpreted as the sequence
5480 of signs for the corresponding indices @code{m}, it must contain 1 or -1, e.g.
5481 @code{zeta(lst(3,4), lst(-1,1))} means
5483 $\zeta(\overline{3},4)$.
5485 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5486 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5487 e.g. @code{lst(0,0,-1,0,1,0,0)}, @code{lst(0,0,-1,2,0,0)} and @code{lst(-3,2,0,0)} are equivalent as
5488 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5489 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5490 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5491 evaluates also for negative integers and positive even integers. For example:
5494 > Li(@{3,1@},@{x,1@});
5497 -zeta(@{3,2@},@{-1,-1@})
5502 It is easy to tell for a given function into which other function it can be rewritten, may
5503 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5504 with negative indices or trailing zeros (the example above gives a hint). Signs can
5505 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5506 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5507 @code{Li} (@code{eval()} already cares for the possible downgrade):
5510 > convert_H_to_Li(@{0,-2,-1,3@},x);
5511 Li(@{3,1,3@},@{-x,1,-1@})
5512 > convert_H_to_Li(@{2,-1,0@},x);
5513 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5516 Every function apart from the multiple polylogarithm @code{Li} can be numerically evaluated for
5517 arbitrary real or complex arguments. @code{Li} only evaluates if for all arguments
5522 $x_1x_2\cdots x_i < 1$ holds.
5528 > evalf(zeta(@{3,1,3,1@}));
5529 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5532 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5533 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5535 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5540 In long expressions this helps a lot with debugging, because you can easily spot
5541 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5542 cancellations of divergencies happen.
5544 Useful publications:
5546 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5547 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5549 @cite{Harmonic Polylogarithms},
5550 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5552 @cite{Special Values of Multiple Polylogarithms},
5553 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5555 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5556 @c node-name, next, previous, up
5557 @section Complex Conjugation
5559 @cindex @code{conjugate()}
5567 returns the complex conjugate of the expression. For all built-in functions and objects the
5568 conjugation gives the expected results:
5572 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5576 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5577 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5578 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5579 // -> -gamma5*gamma~b*gamma~a
5583 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5584 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5585 arguments. This is the default strategy. If you want to define your own functions and want to
5586 change this behavior, you have to supply a specialized conjugation method for your function
5587 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5589 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5590 @c node-name, next, previous, up
5591 @section Solving Linear Systems of Equations
5592 @cindex @code{lsolve()}
5594 The function @code{lsolve()} provides a convenient wrapper around some
5595 matrix operations that comes in handy when a system of linear equations
5599 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5602 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5603 @code{relational}) while @code{symbols} is a @code{lst} of
5604 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5607 It returns the @code{lst} of solutions as an expression. As an example,
5608 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5612 symbol a("a"), b("b"), x("x"), y("y");
5614 eqns = a*x+b*y==3, x-y==b;
5616 cout << lsolve(eqns, vars) << endl;
5617 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5620 When the linear equations @code{eqns} are underdetermined, the solution
5621 will contain one or more tautological entries like @code{x==x},
5622 depending on the rank of the system. When they are overdetermined, the
5623 solution will be an empty @code{lst}. Note the third optional parameter
5624 to @code{lsolve()}: it accepts the same parameters as
5625 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5629 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5630 @c node-name, next, previous, up
5631 @section Input and output of expressions
5634 @subsection Expression output
5636 @cindex output of expressions
5638 Expressions can simply be written to any stream:
5643 ex e = 4.5*I+pow(x,2)*3/2;
5644 cout << e << endl; // prints '4.5*I+3/2*x^2'
5648 The default output format is identical to the @command{ginsh} input syntax and
5649 to that used by most computer algebra systems, but not directly pastable
5650 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5651 is printed as @samp{x^2}).
5653 It is possible to print expressions in a number of different formats with
5654 a set of stream manipulators;
5657 std::ostream & dflt(std::ostream & os);
5658 std::ostream & latex(std::ostream & os);
5659 std::ostream & tree(std::ostream & os);
5660 std::ostream & csrc(std::ostream & os);
5661 std::ostream & csrc_float(std::ostream & os);
5662 std::ostream & csrc_double(std::ostream & os);
5663 std::ostream & csrc_cl_N(std::ostream & os);
5664 std::ostream & index_dimensions(std::ostream & os);
5665 std::ostream & no_index_dimensions(std::ostream & os);
5668 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5669 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5670 @code{print_csrc()} functions, respectively.
5673 All manipulators affect the stream state permanently. To reset the output
5674 format to the default, use the @code{dflt} manipulator:
5678 cout << latex; // all output to cout will be in LaTeX format from now on
5679 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5680 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5681 cout << dflt; // revert to default output format
5682 cout << e << endl; // prints '4.5*I+3/2*x^2'
5686 If you don't want to affect the format of the stream you're working with,
5687 you can output to a temporary @code{ostringstream} like this:
5692 s << latex << e; // format of cout remains unchanged
5693 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5698 @cindex @code{csrc_float}
5699 @cindex @code{csrc_double}
5700 @cindex @code{csrc_cl_N}
5701 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5702 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5703 format that can be directly used in a C or C++ program. The three possible
5704 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5705 classes provided by the CLN library):
5709 cout << "f = " << csrc_float << e << ";\n";
5710 cout << "d = " << csrc_double << e << ";\n";
5711 cout << "n = " << csrc_cl_N << e << ";\n";
5715 The above example will produce (note the @code{x^2} being converted to
5719 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5720 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5721 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5725 The @code{tree} manipulator allows dumping the internal structure of an
5726 expression for debugging purposes:
5737 add, hash=0x0, flags=0x3, nops=2
5738 power, hash=0x0, flags=0x3, nops=2
5739 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5740 2 (numeric), hash=0x6526b0fa, flags=0xf
5741 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5744 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5748 @cindex @code{latex}
5749 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5750 It is rather similar to the default format but provides some braces needed
5751 by LaTeX for delimiting boxes and also converts some common objects to
5752 conventional LaTeX names. It is possible to give symbols a special name for
5753 LaTeX output by supplying it as a second argument to the @code{symbol}
5756 For example, the code snippet
5760 symbol x("x", "\\circ");
5761 ex e = lgamma(x).series(x==0,3);
5762 cout << latex << e << endl;
5769 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5772 @cindex @code{index_dimensions}
5773 @cindex @code{no_index_dimensions}
5774 Index dimensions are normally hidden in the output. To make them visible, use
5775 the @code{index_dimensions} manipulator. The dimensions will be written in
5776 square brackets behind each index value in the default and LaTeX output
5781 symbol x("x"), y("y");
5782 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5783 ex e = indexed(x, mu) * indexed(y, nu);
5786 // prints 'x~mu*y~nu'
5787 cout << index_dimensions << e << endl;
5788 // prints 'x~mu[4]*y~nu[4]'
5789 cout << no_index_dimensions << e << endl;
5790 // prints 'x~mu*y~nu'
5795 @cindex Tree traversal
5796 If you need any fancy special output format, e.g. for interfacing GiNaC
5797 with other algebra systems or for producing code for different
5798 programming languages, you can always traverse the expression tree yourself:
5801 static void my_print(const ex & e)
5803 if (is_a<function>(e))
5804 cout << ex_to<function>(e).get_name();
5806 cout << ex_to<basic>(e).class_name();
5808 size_t n = e.nops();
5810 for (size_t i=0; i<n; i++) @{
5822 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5830 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5831 symbol(y))),numeric(-2)))
5834 If you need an output format that makes it possible to accurately
5835 reconstruct an expression by feeding the output to a suitable parser or
5836 object factory, you should consider storing the expression in an
5837 @code{archive} object and reading the object properties from there.
5838 See the section on archiving for more information.
5841 @subsection Expression input
5842 @cindex input of expressions
5844 GiNaC provides no way to directly read an expression from a stream because
5845 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5846 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5847 @code{y} you defined in your program and there is no way to specify the
5848 desired symbols to the @code{>>} stream input operator.
5850 Instead, GiNaC lets you construct an expression from a string, specifying the
5851 list of symbols to be used:
5855 symbol x("x"), y("y");
5856 ex e("2*x+sin(y)", lst(x, y));
5860 The input syntax is the same as that used by @command{ginsh} and the stream
5861 output operator @code{<<}. The symbols in the string are matched by name to
5862 the symbols in the list and if GiNaC encounters a symbol not specified in
5863 the list it will throw an exception.
5865 With this constructor, it's also easy to implement interactive GiNaC programs:
5870 #include <stdexcept>
5871 #include <ginac/ginac.h>
5872 using namespace std;
5873 using namespace GiNaC;
5880 cout << "Enter an expression containing 'x': ";
5885 cout << "The derivative of " << e << " with respect to x is ";
5886 cout << e.diff(x) << ".\n";
5887 @} catch (exception &p) @{
5888 cerr << p.what() << endl;
5894 @subsection Archiving
5895 @cindex @code{archive} (class)
5898 GiNaC allows creating @dfn{archives} of expressions which can be stored
5899 to or retrieved from files. To create an archive, you declare an object
5900 of class @code{archive} and archive expressions in it, giving each
5901 expression a unique name:
5905 using namespace std;
5906 #include <ginac/ginac.h>
5907 using namespace GiNaC;
5911 symbol x("x"), y("y"), z("z");
5913 ex foo = sin(x + 2*y) + 3*z + 41;
5917 a.archive_ex(foo, "foo");
5918 a.archive_ex(bar, "the second one");
5922 The archive can then be written to a file:
5926 ofstream out("foobar.gar");
5932 The file @file{foobar.gar} contains all information that is needed to
5933 reconstruct the expressions @code{foo} and @code{bar}.
5935 @cindex @command{viewgar}
5936 The tool @command{viewgar} that comes with GiNaC can be used to view
5937 the contents of GiNaC archive files:
5940 $ viewgar foobar.gar
5941 foo = 41+sin(x+2*y)+3*z
5942 the second one = 42+sin(x+2*y)+3*z
5945 The point of writing archive files is of course that they can later be
5951 ifstream in("foobar.gar");
5956 And the stored expressions can be retrieved by their name:
5963 ex ex1 = a2.unarchive_ex(syms, "foo");
5964 ex ex2 = a2.unarchive_ex(syms, "the second one");
5966 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5967 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5968 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5972 Note that you have to supply a list of the symbols which are to be inserted
5973 in the expressions. Symbols in archives are stored by their name only and
5974 if you don't specify which symbols you have, unarchiving the expression will
5975 create new symbols with that name. E.g. if you hadn't included @code{x} in
5976 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5977 have had no effect because the @code{x} in @code{ex1} would have been a
5978 different symbol than the @code{x} which was defined at the beginning of
5979 the program, although both would appear as @samp{x} when printed.
5981 You can also use the information stored in an @code{archive} object to
5982 output expressions in a format suitable for exact reconstruction. The
5983 @code{archive} and @code{archive_node} classes have a couple of member
5984 functions that let you access the stored properties:
5987 static void my_print2(const archive_node & n)
5990 n.find_string("class", class_name);
5991 cout << class_name << "(";
5993 archive_node::propinfovector p;
5994 n.get_properties(p);
5996 size_t num = p.size();
5997 for (size_t i=0; i<num; i++) @{
5998 const string &name = p[i].name;
5999 if (name == "class")
6001 cout << name << "=";
6003 unsigned count = p[i].count;
6007 for (unsigned j=0; j<count; j++) @{
6008 switch (p[i].type) @{
6009 case archive_node::PTYPE_BOOL: @{
6011 n.find_bool(name, x, j);
6012 cout << (x ? "true" : "false");
6015 case archive_node::PTYPE_UNSIGNED: @{
6017 n.find_unsigned(name, x, j);
6021 case archive_node::PTYPE_STRING: @{
6023 n.find_string(name, x, j);
6024 cout << '\"' << x << '\"';
6027 case archive_node::PTYPE_NODE: @{
6028 const archive_node &x = n.find_ex_node(name, j);
6050 ex e = pow(2, x) - y;
6052 my_print2(ar.get_top_node(0)); cout << endl;
6060 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6061 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6062 overall_coeff=numeric(number="0"))
6065 Be warned, however, that the set of properties and their meaning for each
6066 class may change between GiNaC versions.
6069 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6070 @c node-name, next, previous, up
6071 @chapter Extending GiNaC
6073 By reading so far you should have gotten a fairly good understanding of
6074 GiNaC's design patterns. From here on you should start reading the
6075 sources. All we can do now is issue some recommendations how to tackle
6076 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6077 develop some useful extension please don't hesitate to contact the GiNaC
6078 authors---they will happily incorporate them into future versions.
6081 * What does not belong into GiNaC:: What to avoid.
6082 * Symbolic functions:: Implementing symbolic functions.
6083 * Printing:: Adding new output formats.
6084 * Structures:: Defining new algebraic classes (the easy way).
6085 * Adding classes:: Defining new algebraic classes (the hard way).
6089 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6090 @c node-name, next, previous, up
6091 @section What doesn't belong into GiNaC
6093 @cindex @command{ginsh}
6094 First of all, GiNaC's name must be read literally. It is designed to be
6095 a library for use within C++. The tiny @command{ginsh} accompanying
6096 GiNaC makes this even more clear: it doesn't even attempt to provide a
6097 language. There are no loops or conditional expressions in
6098 @command{ginsh}, it is merely a window into the library for the
6099 programmer to test stuff (or to show off). Still, the design of a
6100 complete CAS with a language of its own, graphical capabilities and all
6101 this on top of GiNaC is possible and is without doubt a nice project for
6104 There are many built-in functions in GiNaC that do not know how to
6105 evaluate themselves numerically to a precision declared at runtime
6106 (using @code{Digits}). Some may be evaluated at certain points, but not
6107 generally. This ought to be fixed. However, doing numerical
6108 computations with GiNaC's quite abstract classes is doomed to be
6109 inefficient. For this purpose, the underlying foundation classes
6110 provided by CLN are much better suited.
6113 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6114 @c node-name, next, previous, up
6115 @section Symbolic functions
6117 The easiest and most instructive way to start extending GiNaC is probably to
6118 create your own symbolic functions. These are implemented with the help of
6119 two preprocessor macros:
6121 @cindex @code{DECLARE_FUNCTION}
6122 @cindex @code{REGISTER_FUNCTION}
6124 DECLARE_FUNCTION_<n>P(<name>)
6125 REGISTER_FUNCTION(<name>, <options>)
6128 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6129 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6130 parameters of type @code{ex} and returns a newly constructed GiNaC
6131 @code{function} object that represents your function.
6133 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6134 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6135 set of options that associate the symbolic function with C++ functions you
6136 provide to implement the various methods such as evaluation, derivative,
6137 series expansion etc. They also describe additional attributes the function
6138 might have, such as symmetry and commutation properties, and a name for
6139 LaTeX output. Multiple options are separated by the member access operator
6140 @samp{.} and can be given in an arbitrary order.
6142 (By the way: in case you are worrying about all the macros above we can
6143 assure you that functions are GiNaC's most macro-intense classes. We have
6144 done our best to avoid macros where we can.)
6146 @subsection A minimal example
6148 Here is an example for the implementation of a function with two arguments
6149 that is not further evaluated:
6152 DECLARE_FUNCTION_2P(myfcn)
6154 REGISTER_FUNCTION(myfcn, dummy())
6157 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6158 in algebraic expressions:
6164 ex e = 2*myfcn(42, 1+3*x) - x;
6166 // prints '2*myfcn(42,1+3*x)-x'
6171 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6172 "no options". A function with no options specified merely acts as a kind of
6173 container for its arguments. It is a pure "dummy" function with no associated
6174 logic (which is, however, sometimes perfectly sufficient).
6176 Let's now have a look at the implementation of GiNaC's cosine function for an
6177 example of how to make an "intelligent" function.
6179 @subsection The cosine function
6181 The GiNaC header file @file{inifcns.h} contains the line
6184 DECLARE_FUNCTION_1P(cos)
6187 which declares to all programs using GiNaC that there is a function @samp{cos}
6188 that takes one @code{ex} as an argument. This is all they need to know to use
6189 this function in expressions.
6191 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6192 is its @code{REGISTER_FUNCTION} line:
6195 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6196 evalf_func(cos_evalf).
6197 derivative_func(cos_deriv).
6198 latex_name("\\cos"));
6201 There are four options defined for the cosine function. One of them
6202 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6203 other three indicate the C++ functions in which the "brains" of the cosine
6204 function are defined.
6206 @cindex @code{hold()}
6208 The @code{eval_func()} option specifies the C++ function that implements
6209 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6210 the same number of arguments as the associated symbolic function (one in this
6211 case) and returns the (possibly transformed or in some way simplified)
6212 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6213 of the automatic evaluation process). If no (further) evaluation is to take
6214 place, the @code{eval_func()} function must return the original function
6215 with @code{.hold()}, to avoid a potential infinite recursion. If your
6216 symbolic functions produce a segmentation fault or stack overflow when
6217 using them in expressions, you are probably missing a @code{.hold()}
6220 The @code{eval_func()} function for the cosine looks something like this
6221 (actually, it doesn't look like this at all, but it should give you an idea
6225 static ex cos_eval(const ex & x)
6227 if ("x is a multiple of 2*Pi")
6229 else if ("x is a multiple of Pi")
6231 else if ("x is a multiple of Pi/2")
6235 else if ("x has the form 'acos(y)'")
6237 else if ("x has the form 'asin(y)'")
6242 return cos(x).hold();
6246 This function is called every time the cosine is used in a symbolic expression:
6252 // this calls cos_eval(Pi), and inserts its return value into
6253 // the actual expression
6260 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6261 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6262 symbolic transformation can be done, the unmodified function is returned
6263 with @code{.hold()}.
6265 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6266 The user has to call @code{evalf()} for that. This is implemented in a
6270 static ex cos_evalf(const ex & x)
6272 if (is_a<numeric>(x))
6273 return cos(ex_to<numeric>(x));
6275 return cos(x).hold();
6279 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6280 in this case the @code{cos()} function for @code{numeric} objects, which in
6281 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6282 isn't really needed here, but reminds us that the corresponding @code{eval()}
6283 function would require it in this place.
6285 Differentiation will surely turn up and so we need to tell @code{cos}
6286 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6287 instance, are then handled automatically by @code{basic::diff} and
6291 static ex cos_deriv(const ex & x, unsigned diff_param)
6297 @cindex product rule
6298 The second parameter is obligatory but uninteresting at this point. It
6299 specifies which parameter to differentiate in a partial derivative in
6300 case the function has more than one parameter, and its main application
6301 is for correct handling of the chain rule.
6303 An implementation of the series expansion is not needed for @code{cos()} as
6304 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6305 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6306 the other hand, does have poles and may need to do Laurent expansion:
6309 static ex tan_series(const ex & x, const relational & rel,
6310 int order, unsigned options)
6312 // Find the actual expansion point
6313 const ex x_pt = x.subs(rel);
6315 if ("x_pt is not an odd multiple of Pi/2")
6316 throw do_taylor(); // tell function::series() to do Taylor expansion
6318 // On a pole, expand sin()/cos()
6319 return (sin(x)/cos(x)).series(rel, order+2, options);
6323 The @code{series()} implementation of a function @emph{must} return a
6324 @code{pseries} object, otherwise your code will crash.
6326 @subsection Function options
6328 GiNaC functions understand several more options which are always
6329 specified as @code{.option(params)}. None of them are required, but you
6330 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6331 is a do-nothing option called @code{dummy()} which you can use to define
6332 functions without any special options.
6335 eval_func(<C++ function>)
6336 evalf_func(<C++ function>)
6337 derivative_func(<C++ function>)
6338 series_func(<C++ function>)
6339 conjugate_func(<C++ function>)
6342 These specify the C++ functions that implement symbolic evaluation,
6343 numeric evaluation, partial derivatives, and series expansion, respectively.
6344 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6345 @code{diff()} and @code{series()}.
6347 The @code{eval_func()} function needs to use @code{.hold()} if no further
6348 automatic evaluation is desired or possible.
6350 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6351 expansion, which is correct if there are no poles involved. If the function
6352 has poles in the complex plane, the @code{series_func()} needs to check
6353 whether the expansion point is on a pole and fall back to Taylor expansion
6354 if it isn't. Otherwise, the pole usually needs to be regularized by some
6355 suitable transformation.
6358 latex_name(const string & n)
6361 specifies the LaTeX code that represents the name of the function in LaTeX
6362 output. The default is to put the function name in an @code{\mbox@{@}}.
6365 do_not_evalf_params()
6368 This tells @code{evalf()} to not recursively evaluate the parameters of the
6369 function before calling the @code{evalf_func()}.
6372 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6375 This allows you to explicitly specify the commutation properties of the
6376 function (@xref{Non-commutative objects}, for an explanation of
6377 (non)commutativity in GiNaC). For example, you can use
6378 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6379 GiNaC treat your function like a matrix. By default, functions inherit the
6380 commutation properties of their first argument.
6383 set_symmetry(const symmetry & s)
6386 specifies the symmetry properties of the function with respect to its
6387 arguments. @xref{Indexed objects}, for an explanation of symmetry
6388 specifications. GiNaC will automatically rearrange the arguments of
6389 symmetric functions into a canonical order.
6391 Sometimes you may want to have finer control over how functions are
6392 displayed in the output. For example, the @code{abs()} function prints
6393 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6394 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6398 print_func<C>(<C++ function>)
6401 option which is explained in the next section.
6403 @subsection Functions with a variable number of arguments
6405 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6406 functions with a fixed number of arguments. Sometimes, though, you may need
6407 to have a function that accepts a variable number of expressions. One way to
6408 accomplish this is to pass variable-length lists as arguments. The
6409 @code{Li()} function uses this method for multiple polylogarithms.
6411 It is also possible to define functions that accept a different number of
6412 parameters under the same function name, such as the @code{psi()} function
6413 which can be called either as @code{psi(z)} (the digamma function) or as
6414 @code{psi(n, z)} (polygamma functions). These are actually two different
6415 functions in GiNaC that, however, have the same name. Defining such
6416 functions is not possible with the macros but requires manually fiddling
6417 with GiNaC internals. If you are interested, please consult the GiNaC source
6418 code for the @code{psi()} function (@file{inifcns.h} and
6419 @file{inifcns_gamma.cpp}).
6422 @node Printing, Structures, Symbolic functions, Extending GiNaC
6423 @c node-name, next, previous, up
6424 @section GiNaC's expression output system
6426 GiNaC allows the output of expressions in a variety of different formats
6427 (@pxref{Input/Output}). This section will explain how expression output
6428 is implemented internally, and how to define your own output formats or
6429 change the output format of built-in algebraic objects. You will also want
6430 to read this section if you plan to write your own algebraic classes or
6433 @cindex @code{print_context} (class)
6434 @cindex @code{print_dflt} (class)
6435 @cindex @code{print_latex} (class)
6436 @cindex @code{print_tree} (class)
6437 @cindex @code{print_csrc} (class)
6438 All the different output formats are represented by a hierarchy of classes
6439 rooted in the @code{print_context} class, defined in the @file{print.h}
6444 the default output format
6446 output in LaTeX mathematical mode
6448 a dump of the internal expression structure (for debugging)
6450 the base class for C source output
6451 @item print_csrc_float
6452 C source output using the @code{float} type
6453 @item print_csrc_double
6454 C source output using the @code{double} type
6455 @item print_csrc_cl_N
6456 C source output using CLN types
6459 The @code{print_context} base class provides two public data members:
6471 @code{s} is a reference to the stream to output to, while @code{options}
6472 holds flags and modifiers. Currently, there is only one flag defined:
6473 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6474 to print the index dimension which is normally hidden.
6476 When you write something like @code{std::cout << e}, where @code{e} is
6477 an object of class @code{ex}, GiNaC will construct an appropriate
6478 @code{print_context} object (of a class depending on the selected output
6479 format), fill in the @code{s} and @code{options} members, and call
6481 @cindex @code{print()}
6483 void ex::print(const print_context & c, unsigned level = 0) const;
6486 which in turn forwards the call to the @code{print()} method of the
6487 top-level algebraic object contained in the expression.
6489 Unlike other methods, GiNaC classes don't usually override their
6490 @code{print()} method to implement expression output. Instead, the default
6491 implementation @code{basic::print(c, level)} performs a run-time double
6492 dispatch to a function selected by the dynamic type of the object and the
6493 passed @code{print_context}. To this end, GiNaC maintains a separate method
6494 table for each class, similar to the virtual function table used for ordinary
6495 (single) virtual function dispatch.
6497 The method table contains one slot for each possible @code{print_context}
6498 type, indexed by the (internally assigned) serial number of the type. Slots
6499 may be empty, in which case GiNaC will retry the method lookup with the
6500 @code{print_context} object's parent class, possibly repeating the process
6501 until it reaches the @code{print_context} base class. If there's still no
6502 method defined, the method table of the algebraic object's parent class
6503 is consulted, and so on, until a matching method is found (eventually it
6504 will reach the combination @code{basic/print_context}, which prints the
6505 object's class name enclosed in square brackets).
6507 You can think of the print methods of all the different classes and output
6508 formats as being arranged in a two-dimensional matrix with one axis listing
6509 the algebraic classes and the other axis listing the @code{print_context}
6512 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6513 to implement printing, but then they won't get any of the benefits of the
6514 double dispatch mechanism (such as the ability for derived classes to
6515 inherit only certain print methods from its parent, or the replacement of
6516 methods at run-time).
6518 @subsection Print methods for classes
6520 The method table for a class is set up either in the definition of the class,
6521 by passing the appropriate @code{print_func<C>()} option to
6522 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6523 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6524 can also be used to override existing methods dynamically.
6526 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6527 be a member function of the class (or one of its parent classes), a static
6528 member function, or an ordinary (global) C++ function. The @code{C} template
6529 parameter specifies the appropriate @code{print_context} type for which the
6530 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6531 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6532 the class is the one being implemented by
6533 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6535 For print methods that are member functions, their first argument must be of
6536 a type convertible to a @code{const C &}, and the second argument must be an
6539 For static members and global functions, the first argument must be of a type
6540 convertible to a @code{const T &}, the second argument must be of a type
6541 convertible to a @code{const C &}, and the third argument must be an
6542 @code{unsigned}. A global function will, of course, not have access to
6543 private and protected members of @code{T}.
6545 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6546 and @code{basic::print()}) is used for proper parenthesizing of the output
6547 (and by @code{print_tree} for proper indentation). It can be used for similar
6548 purposes if you write your own output formats.
6550 The explanations given above may seem complicated, but in practice it's
6551 really simple, as shown in the following example. Suppose that we want to
6552 display exponents in LaTeX output not as superscripts but with little
6553 upwards-pointing arrows. This can be achieved in the following way:
6556 void my_print_power_as_latex(const power & p,
6557 const print_latex & c,
6560 // get the precedence of the 'power' class
6561 unsigned power_prec = p.precedence();
6563 // if the parent operator has the same or a higher precedence
6564 // we need parentheses around the power
6565 if (level >= power_prec)
6568 // print the basis and exponent, each enclosed in braces, and
6569 // separated by an uparrow
6571 p.op(0).print(c, power_prec);
6572 c.s << "@}\\uparrow@{";
6573 p.op(1).print(c, power_prec);
6576 // don't forget the closing parenthesis
6577 if (level >= power_prec)
6583 // a sample expression
6584 symbol x("x"), y("y");
6585 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6587 // switch to LaTeX mode
6590 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6593 // now we replace the method for the LaTeX output of powers with
6595 set_print_func<power, print_latex>(my_print_power_as_latex);
6597 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6607 The first argument of @code{my_print_power_as_latex} could also have been
6608 a @code{const basic &}, the second one a @code{const print_context &}.
6611 The above code depends on @code{mul} objects converting their operands to
6612 @code{power} objects for the purpose of printing.
6615 The output of products including negative powers as fractions is also
6616 controlled by the @code{mul} class.
6619 The @code{power/print_latex} method provided by GiNaC prints square roots
6620 using @code{\sqrt}, but the above code doesn't.
6624 It's not possible to restore a method table entry to its previous or default
6625 value. Once you have called @code{set_print_func()}, you can only override
6626 it with another call to @code{set_print_func()}, but you can't easily go back
6627 to the default behavior again (you can, of course, dig around in the GiNaC
6628 sources, find the method that is installed at startup
6629 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6630 one; that is, after you circumvent the C++ member access control@dots{}).
6632 @subsection Print methods for functions
6634 Symbolic functions employ a print method dispatch mechanism similar to the
6635 one used for classes. The methods are specified with @code{print_func<C>()}
6636 function options. If you don't specify any special print methods, the function
6637 will be printed with its name (or LaTeX name, if supplied), followed by a
6638 comma-separated list of arguments enclosed in parentheses.
6640 For example, this is what GiNaC's @samp{abs()} function is defined like:
6643 static ex abs_eval(const ex & arg) @{ ... @}
6644 static ex abs_evalf(const ex & arg) @{ ... @}
6646 static void abs_print_latex(const ex & arg, const print_context & c)
6648 c.s << "@{|"; arg.print(c); c.s << "|@}";
6651 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6653 c.s << "fabs("; arg.print(c); c.s << ")";
6656 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6657 evalf_func(abs_evalf).
6658 print_func<print_latex>(abs_print_latex).
6659 print_func<print_csrc_float>(abs_print_csrc_float).
6660 print_func<print_csrc_double>(abs_print_csrc_float));
6663 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6664 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6666 There is currently no equivalent of @code{set_print_func()} for functions.
6668 @subsection Adding new output formats
6670 Creating a new output format involves subclassing @code{print_context},
6671 which is somewhat similar to adding a new algebraic class
6672 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6673 that needs to go into the class definition, and a corresponding macro
6674 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6675 Every @code{print_context} class needs to provide a default constructor
6676 and a constructor from an @code{std::ostream} and an @code{unsigned}
6679 Here is an example for a user-defined @code{print_context} class:
6682 class print_myformat : public print_dflt
6684 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6686 print_myformat(std::ostream & os, unsigned opt = 0)
6687 : print_dflt(os, opt) @{@}
6690 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6692 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6695 That's all there is to it. None of the actual expression output logic is
6696 implemented in this class. It merely serves as a selector for choosing
6697 a particular format. The algorithms for printing expressions in the new
6698 format are implemented as print methods, as described above.
6700 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6701 exactly like GiNaC's default output format:
6706 ex e = pow(x, 2) + 1;
6708 // this prints "1+x^2"
6711 // this also prints "1+x^2"
6712 e.print(print_myformat()); cout << endl;
6718 To fill @code{print_myformat} with life, we need to supply appropriate
6719 print methods with @code{set_print_func()}, like this:
6722 // This prints powers with '**' instead of '^'. See the LaTeX output
6723 // example above for explanations.
6724 void print_power_as_myformat(const power & p,
6725 const print_myformat & c,
6728 unsigned power_prec = p.precedence();
6729 if (level >= power_prec)
6731 p.op(0).print(c, power_prec);
6733 p.op(1).print(c, power_prec);
6734 if (level >= power_prec)
6740 // install a new print method for power objects
6741 set_print_func<power, print_myformat>(print_power_as_myformat);
6743 // now this prints "1+x**2"
6744 e.print(print_myformat()); cout << endl;
6746 // but the default format is still "1+x^2"
6752 @node Structures, Adding classes, Printing, Extending GiNaC
6753 @c node-name, next, previous, up
6756 If you are doing some very specialized things with GiNaC, or if you just
6757 need some more organized way to store data in your expressions instead of
6758 anonymous lists, you may want to implement your own algebraic classes.
6759 ('algebraic class' means any class directly or indirectly derived from
6760 @code{basic} that can be used in GiNaC expressions).
6762 GiNaC offers two ways of accomplishing this: either by using the
6763 @code{structure<T>} template class, or by rolling your own class from
6764 scratch. This section will discuss the @code{structure<T>} template which
6765 is easier to use but more limited, while the implementation of custom
6766 GiNaC classes is the topic of the next section. However, you may want to
6767 read both sections because many common concepts and member functions are
6768 shared by both concepts, and it will also allow you to decide which approach
6769 is most suited to your needs.
6771 The @code{structure<T>} template, defined in the GiNaC header file
6772 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6773 or @code{class}) into a GiNaC object that can be used in expressions.
6775 @subsection Example: scalar products
6777 Let's suppose that we need a way to handle some kind of abstract scalar
6778 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6779 product class have to store their left and right operands, which can in turn
6780 be arbitrary expressions. Here is a possible way to represent such a
6781 product in a C++ @code{struct}:
6785 using namespace std;
6787 #include <ginac/ginac.h>
6788 using namespace GiNaC;
6794 sprod_s(ex l, ex r) : left(l), right(r) @{@}
6798 The default constructor is required. Now, to make a GiNaC class out of this
6799 data structure, we need only one line:
6802 typedef structure<sprod_s> sprod;
6805 That's it. This line constructs an algebraic class @code{sprod} which
6806 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
6807 expressions like any other GiNaC class:
6811 symbol a("a"), b("b");
6812 ex e = sprod(sprod_s(a, b));
6816 Note the difference between @code{sprod} which is the algebraic class, and
6817 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
6818 and @code{right} data members. As shown above, an @code{sprod} can be
6819 constructed from an @code{sprod_s} object.
6821 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
6822 you could define a little wrapper function like this:
6825 inline ex make_sprod(ex left, ex right)
6827 return sprod(sprod_s(left, right));
6831 The @code{sprod_s} object contained in @code{sprod} can be accessed with
6832 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
6833 @code{get_struct()}:
6837 cout << ex_to<sprod>(e)->left << endl;
6839 cout << ex_to<sprod>(e).get_struct().right << endl;
6844 You only have read access to the members of @code{sprod_s}.
6846 The type definition of @code{sprod} is enough to write your own algorithms
6847 that deal with scalar products, for example:
6852 if (is_a<sprod>(p)) @{
6853 const sprod_s & sp = ex_to<sprod>(p).get_struct();
6854 return make_sprod(sp.right, sp.left);
6865 @subsection Structure output
6867 While the @code{sprod} type is useable it still leaves something to be
6868 desired, most notably proper output:
6873 // -> [structure object]
6877 By default, any structure types you define will be printed as
6878 @samp{[structure object]}. To override this you can either specialize the
6879 template's @code{print()} member function, or specify print methods with
6880 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
6881 it's not possible to supply class options like @code{print_func<>()} to
6882 structures, so for a self-contained structure type you need to resort to
6883 overriding the @code{print()} function, which is also what we will do here.
6885 The member functions of GiNaC classes are described in more detail in the
6886 next section, but it shouldn't be hard to figure out what's going on here:
6889 void sprod::print(const print_context & c, unsigned level) const
6891 // tree debug output handled by superclass
6892 if (is_a<print_tree>(c))
6893 inherited::print(c, level);
6895 // get the contained sprod_s object
6896 const sprod_s & sp = get_struct();
6898 // print_context::s is a reference to an ostream
6899 c.s << "<" << sp.left << "|" << sp.right << ">";
6903 Now we can print expressions containing scalar products:
6909 cout << swap_sprod(e) << endl;
6914 @subsection Comparing structures
6916 The @code{sprod} class defined so far still has one important drawback: all
6917 scalar products are treated as being equal because GiNaC doesn't know how to
6918 compare objects of type @code{sprod_s}. This can lead to some confusing
6919 and undesired behavior:
6923 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6925 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6926 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
6930 To remedy this, we first need to define the operators @code{==} and @code{<}
6931 for objects of type @code{sprod_s}:
6934 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
6936 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
6939 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
6941 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
6945 The ordering established by the @code{<} operator doesn't have to make any
6946 algebraic sense, but it needs to be well defined. Note that we can't use
6947 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
6948 in the implementation of these operators because they would construct
6949 GiNaC @code{relational} objects which in the case of @code{<} do not
6950 establish a well defined ordering (for arbitrary expressions, GiNaC can't
6951 decide which one is algebraically 'less').
6953 Next, we need to change our definition of the @code{sprod} type to let
6954 GiNaC know that an ordering relation exists for the embedded objects:
6957 typedef structure<sprod_s, compare_std_less> sprod;
6960 @code{sprod} objects then behave as expected:
6964 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6965 // -> <a|b>-<a^2|b^2>
6966 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6967 // -> <a|b>+<a^2|b^2>
6968 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
6970 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
6975 The @code{compare_std_less} policy parameter tells GiNaC to use the
6976 @code{std::less} and @code{std::equal_to} functors to compare objects of
6977 type @code{sprod_s}. By default, these functors forward their work to the
6978 standard @code{<} and @code{==} operators, which we have overloaded.
6979 Alternatively, we could have specialized @code{std::less} and
6980 @code{std::equal_to} for class @code{sprod_s}.
6982 GiNaC provides two other comparison policies for @code{structure<T>}
6983 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
6984 which does a bit-wise comparison of the contained @code{T} objects.
6985 This should be used with extreme care because it only works reliably with
6986 built-in integral types, and it also compares any padding (filler bytes of
6987 undefined value) that the @code{T} class might have.
6989 @subsection Subexpressions
6991 Our scalar product class has two subexpressions: the left and right
6992 operands. It might be a good idea to make them accessible via the standard
6993 @code{nops()} and @code{op()} methods:
6996 size_t sprod::nops() const
7001 ex sprod::op(size_t i) const
7005 return get_struct().left;
7007 return get_struct().right;
7009 throw std::range_error("sprod::op(): no such operand");
7014 Implementing @code{nops()} and @code{op()} for container types such as
7015 @code{sprod} has two other nice side effects:
7019 @code{has()} works as expected
7021 GiNaC generates better hash keys for the objects (the default implementation
7022 of @code{calchash()} takes subexpressions into account)
7025 @cindex @code{let_op()}
7026 There is a non-const variant of @code{op()} called @code{let_op()} that
7027 allows replacing subexpressions:
7030 ex & sprod::let_op(size_t i)
7032 // every non-const member function must call this
7033 ensure_if_modifiable();
7037 return get_struct().left;
7039 return get_struct().right;
7041 throw std::range_error("sprod::let_op(): no such operand");
7046 Once we have provided @code{let_op()} we also get @code{subs()} and
7047 @code{map()} for free. In fact, every container class that returns a non-null
7048 @code{nops()} value must either implement @code{let_op()} or provide custom
7049 implementations of @code{subs()} and @code{map()}.
7051 In turn, the availability of @code{map()} enables the recursive behavior of a
7052 couple of other default method implementations, in particular @code{evalf()},
7053 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7054 we probably want to provide our own version of @code{expand()} for scalar
7055 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7056 This is left as an exercise for the reader.
7058 The @code{structure<T>} template defines many more member functions that
7059 you can override by specialization to customize the behavior of your
7060 structures. You are referred to the next section for a description of
7061 some of these (especially @code{eval()}). There is, however, one topic
7062 that shall be addressed here, as it demonstrates one peculiarity of the
7063 @code{structure<T>} template: archiving.
7065 @subsection Archiving structures
7067 If you don't know how the archiving of GiNaC objects is implemented, you
7068 should first read the next section and then come back here. You're back?
7071 To implement archiving for structures it is not enough to provide
7072 specializations for the @code{archive()} member function and the
7073 unarchiving constructor (the @code{unarchive()} function has a default
7074 implementation). You also need to provide a unique name (as a string literal)
7075 for each structure type you define. This is because in GiNaC archives,
7076 the class of an object is stored as a string, the class name.
7078 By default, this class name (as returned by the @code{class_name()} member
7079 function) is @samp{structure} for all structure classes. This works as long
7080 as you have only defined one structure type, but if you use two or more you
7081 need to provide a different name for each by specializing the
7082 @code{get_class_name()} member function. Here is a sample implementation
7083 for enabling archiving of the scalar product type defined above:
7086 const char *sprod::get_class_name() @{ return "sprod"; @}
7088 void sprod::archive(archive_node & n) const
7090 inherited::archive(n);
7091 n.add_ex("left", get_struct().left);
7092 n.add_ex("right", get_struct().right);
7095 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7097 n.find_ex("left", get_struct().left, sym_lst);
7098 n.find_ex("right", get_struct().right, sym_lst);
7102 Note that the unarchiving constructor is @code{sprod::structure} and not
7103 @code{sprod::sprod}, and that we don't need to supply an
7104 @code{sprod::unarchive()} function.
7107 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7108 @c node-name, next, previous, up
7109 @section Adding classes
7111 The @code{structure<T>} template provides an way to extend GiNaC with custom
7112 algebraic classes that is easy to use but has its limitations, the most
7113 severe of which being that you can't add any new member functions to
7114 structures. To be able to do this, you need to write a new class definition
7117 This section will explain how to implement new algebraic classes in GiNaC by
7118 giving the example of a simple 'string' class. After reading this section
7119 you will know how to properly declare a GiNaC class and what the minimum
7120 required member functions are that you have to implement. We only cover the
7121 implementation of a 'leaf' class here (i.e. one that doesn't contain
7122 subexpressions). Creating a container class like, for example, a class
7123 representing tensor products is more involved but this section should give
7124 you enough information so you can consult the source to GiNaC's predefined
7125 classes if you want to implement something more complicated.
7127 @subsection GiNaC's run-time type information system
7129 @cindex hierarchy of classes
7131 All algebraic classes (that is, all classes that can appear in expressions)
7132 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7133 @code{basic *} (which is essentially what an @code{ex} is) represents a
7134 generic pointer to an algebraic class. Occasionally it is necessary to find
7135 out what the class of an object pointed to by a @code{basic *} really is.
7136 Also, for the unarchiving of expressions it must be possible to find the
7137 @code{unarchive()} function of a class given the class name (as a string). A
7138 system that provides this kind of information is called a run-time type
7139 information (RTTI) system. The C++ language provides such a thing (see the
7140 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7141 implements its own, simpler RTTI.
7143 The RTTI in GiNaC is based on two mechanisms:
7148 The @code{basic} class declares a member variable @code{tinfo_key} which
7149 holds an unsigned integer that identifies the object's class. These numbers
7150 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7151 classes. They all start with @code{TINFO_}.
7154 By means of some clever tricks with static members, GiNaC maintains a list
7155 of information for all classes derived from @code{basic}. The information
7156 available includes the class names, the @code{tinfo_key}s, and pointers
7157 to the unarchiving functions. This class registry is defined in the
7158 @file{registrar.h} header file.
7162 The disadvantage of this proprietary RTTI implementation is that there's
7163 a little more to do when implementing new classes (C++'s RTTI works more
7164 or less automatically) but don't worry, most of the work is simplified by
7167 @subsection A minimalistic example
7169 Now we will start implementing a new class @code{mystring} that allows
7170 placing character strings in algebraic expressions (this is not very useful,
7171 but it's just an example). This class will be a direct subclass of
7172 @code{basic}. You can use this sample implementation as a starting point
7173 for your own classes.
7175 The code snippets given here assume that you have included some header files
7181 #include <stdexcept>
7182 using namespace std;
7184 #include <ginac/ginac.h>
7185 using namespace GiNaC;
7188 The first thing we have to do is to define a @code{tinfo_key} for our new
7189 class. This can be any arbitrary unsigned number that is not already taken
7190 by one of the existing classes but it's better to come up with something
7191 that is unlikely to clash with keys that might be added in the future. The
7192 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7193 which is not a requirement but we are going to stick with this scheme:
7196 const unsigned TINFO_mystring = 0x42420001U;
7199 Now we can write down the class declaration. The class stores a C++
7200 @code{string} and the user shall be able to construct a @code{mystring}
7201 object from a C or C++ string:
7204 class mystring : public basic
7206 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7209 mystring(const string &s);
7210 mystring(const char *s);
7216 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7219 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7220 macros are defined in @file{registrar.h}. They take the name of the class
7221 and its direct superclass as arguments and insert all required declarations
7222 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7223 the first line after the opening brace of the class definition. The
7224 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7225 source (at global scope, of course, not inside a function).
7227 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7228 declarations of the default constructor and a couple of other functions that
7229 are required. It also defines a type @code{inherited} which refers to the
7230 superclass so you don't have to modify your code every time you shuffle around
7231 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7232 class with the GiNaC RTTI (there is also a
7233 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7234 options for the class, and which we will be using instead in a few minutes).
7236 Now there are seven member functions we have to implement to get a working
7242 @code{mystring()}, the default constructor.
7245 @code{void archive(archive_node &n)}, the archiving function. This stores all
7246 information needed to reconstruct an object of this class inside an
7247 @code{archive_node}.
7250 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7251 constructor. This constructs an instance of the class from the information
7252 found in an @code{archive_node}.
7255 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7256 unarchiving function. It constructs a new instance by calling the unarchiving
7260 @cindex @code{compare_same_type()}
7261 @code{int compare_same_type(const basic &other)}, which is used internally
7262 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7263 -1, depending on the relative order of this object and the @code{other}
7264 object. If it returns 0, the objects are considered equal.
7265 @strong{Note:} This has nothing to do with the (numeric) ordering
7266 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7267 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7268 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7269 must provide a @code{compare_same_type()} function, even those representing
7270 objects for which no reasonable algebraic ordering relationship can be
7274 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7275 which are the two constructors we declared.
7279 Let's proceed step-by-step. The default constructor looks like this:
7282 mystring::mystring() : inherited(TINFO_mystring) @{@}
7285 The golden rule is that in all constructors you have to set the
7286 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7287 it will be set by the constructor of the superclass and all hell will break
7288 loose in the RTTI. For your convenience, the @code{basic} class provides
7289 a constructor that takes a @code{tinfo_key} value, which we are using here
7290 (remember that in our case @code{inherited == basic}). If the superclass
7291 didn't have such a constructor, we would have to set the @code{tinfo_key}
7292 to the right value manually.
7294 In the default constructor you should set all other member variables to
7295 reasonable default values (we don't need that here since our @code{str}
7296 member gets set to an empty string automatically).
7298 Next are the three functions for archiving. You have to implement them even
7299 if you don't plan to use archives, but the minimum required implementation
7300 is really simple. First, the archiving function:
7303 void mystring::archive(archive_node &n) const
7305 inherited::archive(n);
7306 n.add_string("string", str);
7310 The only thing that is really required is calling the @code{archive()}
7311 function of the superclass. Optionally, you can store all information you
7312 deem necessary for representing the object into the passed
7313 @code{archive_node}. We are just storing our string here. For more
7314 information on how the archiving works, consult the @file{archive.h} header
7317 The unarchiving constructor is basically the inverse of the archiving
7321 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7323 n.find_string("string", str);
7327 If you don't need archiving, just leave this function empty (but you must
7328 invoke the unarchiving constructor of the superclass). Note that we don't
7329 have to set the @code{tinfo_key} here because it is done automatically
7330 by the unarchiving constructor of the @code{basic} class.
7332 Finally, the unarchiving function:
7335 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7337 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7341 You don't have to understand how exactly this works. Just copy these
7342 four lines into your code literally (replacing the class name, of
7343 course). It calls the unarchiving constructor of the class and unless
7344 you are doing something very special (like matching @code{archive_node}s
7345 to global objects) you don't need a different implementation. For those
7346 who are interested: setting the @code{dynallocated} flag puts the object
7347 under the control of GiNaC's garbage collection. It will get deleted
7348 automatically once it is no longer referenced.
7350 Our @code{compare_same_type()} function uses a provided function to compare
7354 int mystring::compare_same_type(const basic &other) const
7356 const mystring &o = static_cast<const mystring &>(other);
7357 int cmpval = str.compare(o.str);
7360 else if (cmpval < 0)
7367 Although this function takes a @code{basic &}, it will always be a reference
7368 to an object of exactly the same class (objects of different classes are not
7369 comparable), so the cast is safe. If this function returns 0, the two objects
7370 are considered equal (in the sense that @math{A-B=0}), so you should compare
7371 all relevant member variables.
7373 Now the only thing missing is our two new constructors:
7376 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7377 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7380 No surprises here. We set the @code{str} member from the argument and
7381 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7383 That's it! We now have a minimal working GiNaC class that can store
7384 strings in algebraic expressions. Let's confirm that the RTTI works:
7387 ex e = mystring("Hello, world!");
7388 cout << is_a<mystring>(e) << endl;
7391 cout << e.bp->class_name() << endl;
7395 Obviously it does. Let's see what the expression @code{e} looks like:
7399 // -> [mystring object]
7402 Hm, not exactly what we expect, but of course the @code{mystring} class
7403 doesn't yet know how to print itself. This can be done either by implementing
7404 the @code{print()} member function, or, preferably, by specifying a
7405 @code{print_func<>()} class option. Let's say that we want to print the string
7406 surrounded by double quotes:
7409 class mystring : public basic
7413 void do_print(const print_context &c, unsigned level = 0) const;
7417 void mystring::do_print(const print_context &c, unsigned level) const
7419 // print_context::s is a reference to an ostream
7420 c.s << '\"' << str << '\"';
7424 The @code{level} argument is only required for container classes to
7425 correctly parenthesize the output.
7427 Now we need to tell GiNaC that @code{mystring} objects should use the
7428 @code{do_print()} member function for printing themselves. For this, we
7432 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7438 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7439 print_func<print_context>(&mystring::do_print))
7442 Let's try again to print the expression:
7446 // -> "Hello, world!"
7449 Much better. If we wanted to have @code{mystring} objects displayed in a
7450 different way depending on the output format (default, LaTeX, etc.), we
7451 would have supplied multiple @code{print_func<>()} options with different
7452 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7453 separated by dots. This is similar to the way options are specified for
7454 symbolic functions. @xref{Printing}, for a more in-depth description of the
7455 way expression output is implemented in GiNaC.
7457 The @code{mystring} class can be used in arbitrary expressions:
7460 e += mystring("GiNaC rulez");
7462 // -> "GiNaC rulez"+"Hello, world!"
7465 (GiNaC's automatic term reordering is in effect here), or even
7468 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7470 // -> "One string"^(2*sin(-"Another string"+Pi))
7473 Whether this makes sense is debatable but remember that this is only an
7474 example. At least it allows you to implement your own symbolic algorithms
7477 Note that GiNaC's algebraic rules remain unchanged:
7480 e = mystring("Wow") * mystring("Wow");
7484 e = pow(mystring("First")-mystring("Second"), 2);
7485 cout << e.expand() << endl;
7486 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7489 There's no way to, for example, make GiNaC's @code{add} class perform string
7490 concatenation. You would have to implement this yourself.
7492 @subsection Automatic evaluation
7495 @cindex @code{eval()}
7496 @cindex @code{hold()}
7497 When dealing with objects that are just a little more complicated than the
7498 simple string objects we have implemented, chances are that you will want to
7499 have some automatic simplifications or canonicalizations performed on them.
7500 This is done in the evaluation member function @code{eval()}. Let's say that
7501 we wanted all strings automatically converted to lowercase with
7502 non-alphabetic characters stripped, and empty strings removed:
7505 class mystring : public basic
7509 ex eval(int level = 0) const;
7513 ex mystring::eval(int level) const
7516 for (int i=0; i<str.length(); i++) @{
7518 if (c >= 'A' && c <= 'Z')
7519 new_str += tolower(c);
7520 else if (c >= 'a' && c <= 'z')
7524 if (new_str.length() == 0)
7527 return mystring(new_str).hold();
7531 The @code{level} argument is used to limit the recursion depth of the
7532 evaluation. We don't have any subexpressions in the @code{mystring}
7533 class so we are not concerned with this. If we had, we would call the
7534 @code{eval()} functions of the subexpressions with @code{level - 1} as
7535 the argument if @code{level != 1}. The @code{hold()} member function
7536 sets a flag in the object that prevents further evaluation. Otherwise
7537 we might end up in an endless loop. When you want to return the object
7538 unmodified, use @code{return this->hold();}.
7540 Let's confirm that it works:
7543 ex e = mystring("Hello, world!") + mystring("!?#");
7547 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7552 @subsection Optional member functions
7554 We have implemented only a small set of member functions to make the class
7555 work in the GiNaC framework. There are two functions that are not strictly
7556 required but will make operations with objects of the class more efficient:
7558 @cindex @code{calchash()}
7559 @cindex @code{is_equal_same_type()}
7561 unsigned calchash() const;
7562 bool is_equal_same_type(const basic &other) const;
7565 The @code{calchash()} method returns an @code{unsigned} hash value for the
7566 object which will allow GiNaC to compare and canonicalize expressions much
7567 more efficiently. You should consult the implementation of some of the built-in
7568 GiNaC classes for examples of hash functions. The default implementation of
7569 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7570 class and all subexpressions that are accessible via @code{op()}.
7572 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7573 tests for equality without establishing an ordering relation, which is often
7574 faster. The default implementation of @code{is_equal_same_type()} just calls
7575 @code{compare_same_type()} and tests its result for zero.
7577 @subsection Other member functions
7579 For a real algebraic class, there are probably some more functions that you
7580 might want to provide:
7583 bool info(unsigned inf) const;
7584 ex evalf(int level = 0) const;
7585 ex series(const relational & r, int order, unsigned options = 0) const;
7586 ex derivative(const symbol & s) const;
7589 If your class stores sub-expressions (see the scalar product example in the
7590 previous section) you will probably want to override
7592 @cindex @code{let_op()}
7595 ex op(size_t i) const;
7596 ex & let_op(size_t i);
7597 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7598 ex map(map_function & f) const;
7601 @code{let_op()} is a variant of @code{op()} that allows write access. The
7602 default implementations of @code{subs()} and @code{map()} use it, so you have
7603 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7605 You can, of course, also add your own new member functions. Remember
7606 that the RTTI may be used to get information about what kinds of objects
7607 you are dealing with (the position in the class hierarchy) and that you
7608 can always extract the bare object from an @code{ex} by stripping the
7609 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7610 should become a need.
7612 That's it. May the source be with you!
7615 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7616 @c node-name, next, previous, up
7617 @chapter A Comparison With Other CAS
7620 This chapter will give you some information on how GiNaC compares to
7621 other, traditional Computer Algebra Systems, like @emph{Maple},
7622 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7623 disadvantages over these systems.
7626 * Advantages:: Strengths of the GiNaC approach.
7627 * Disadvantages:: Weaknesses of the GiNaC approach.
7628 * Why C++?:: Attractiveness of C++.
7631 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7632 @c node-name, next, previous, up
7635 GiNaC has several advantages over traditional Computer
7636 Algebra Systems, like
7641 familiar language: all common CAS implement their own proprietary
7642 grammar which you have to learn first (and maybe learn again when your
7643 vendor decides to `enhance' it). With GiNaC you can write your program
7644 in common C++, which is standardized.
7648 structured data types: you can build up structured data types using
7649 @code{struct}s or @code{class}es together with STL features instead of
7650 using unnamed lists of lists of lists.
7653 strongly typed: in CAS, you usually have only one kind of variables
7654 which can hold contents of an arbitrary type. This 4GL like feature is
7655 nice for novice programmers, but dangerous.
7658 development tools: powerful development tools exist for C++, like fancy
7659 editors (e.g. with automatic indentation and syntax highlighting),
7660 debuggers, visualization tools, documentation generators@dots{}
7663 modularization: C++ programs can easily be split into modules by
7664 separating interface and implementation.
7667 price: GiNaC is distributed under the GNU Public License which means
7668 that it is free and available with source code. And there are excellent
7669 C++-compilers for free, too.
7672 extendable: you can add your own classes to GiNaC, thus extending it on
7673 a very low level. Compare this to a traditional CAS that you can
7674 usually only extend on a high level by writing in the language defined
7675 by the parser. In particular, it turns out to be almost impossible to
7676 fix bugs in a traditional system.
7679 multiple interfaces: Though real GiNaC programs have to be written in
7680 some editor, then be compiled, linked and executed, there are more ways
7681 to work with the GiNaC engine. Many people want to play with
7682 expressions interactively, as in traditional CASs. Currently, two such
7683 windows into GiNaC have been implemented and many more are possible: the
7684 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7685 types to a command line and second, as a more consistent approach, an
7686 interactive interface to the Cint C++ interpreter has been put together
7687 (called GiNaC-cint) that allows an interactive scripting interface
7688 consistent with the C++ language. It is available from the usual GiNaC
7692 seamless integration: it is somewhere between difficult and impossible
7693 to call CAS functions from within a program written in C++ or any other
7694 programming language and vice versa. With GiNaC, your symbolic routines
7695 are part of your program. You can easily call third party libraries,
7696 e.g. for numerical evaluation or graphical interaction. All other
7697 approaches are much more cumbersome: they range from simply ignoring the
7698 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7699 system (i.e. @emph{Yacas}).
7702 efficiency: often large parts of a program do not need symbolic
7703 calculations at all. Why use large integers for loop variables or
7704 arbitrary precision arithmetics where @code{int} and @code{double} are
7705 sufficient? For pure symbolic applications, GiNaC is comparable in
7706 speed with other CAS.
7711 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7712 @c node-name, next, previous, up
7713 @section Disadvantages
7715 Of course it also has some disadvantages:
7720 advanced features: GiNaC cannot compete with a program like
7721 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7722 which grows since 1981 by the work of dozens of programmers, with
7723 respect to mathematical features. Integration, factorization,
7724 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7725 not planned for the near future).
7728 portability: While the GiNaC library itself is designed to avoid any
7729 platform dependent features (it should compile on any ANSI compliant C++
7730 compiler), the currently used version of the CLN library (fast large
7731 integer and arbitrary precision arithmetics) can only by compiled
7732 without hassle on systems with the C++ compiler from the GNU Compiler
7733 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7734 macros to let the compiler gather all static initializations, which
7735 works for GNU C++ only. Feel free to contact the authors in case you
7736 really believe that you need to use a different compiler. We have
7737 occasionally used other compilers and may be able to give you advice.}
7738 GiNaC uses recent language features like explicit constructors, mutable
7739 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7740 literally. Recent GCC versions starting at 2.95.3, although itself not
7741 yet ANSI compliant, support all needed features.
7746 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7747 @c node-name, next, previous, up
7750 Why did we choose to implement GiNaC in C++ instead of Java or any other
7751 language? C++ is not perfect: type checking is not strict (casting is
7752 possible), separation between interface and implementation is not
7753 complete, object oriented design is not enforced. The main reason is
7754 the often scolded feature of operator overloading in C++. While it may
7755 be true that operating on classes with a @code{+} operator is rarely
7756 meaningful, it is perfectly suited for algebraic expressions. Writing
7757 @math{3x+5y} as @code{3*x+5*y} instead of
7758 @code{x.times(3).plus(y.times(5))} looks much more natural.
7759 Furthermore, the main developers are more familiar with C++ than with
7760 any other programming language.
7763 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7764 @c node-name, next, previous, up
7765 @appendix Internal Structures
7768 * Expressions are reference counted::
7769 * Internal representation of products and sums::
7772 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7773 @c node-name, next, previous, up
7774 @appendixsection Expressions are reference counted
7776 @cindex reference counting
7777 @cindex copy-on-write
7778 @cindex garbage collection
7779 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7780 where the counter belongs to the algebraic objects derived from class
7781 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7782 which @code{ex} contains an instance. If you understood that, you can safely
7783 skip the rest of this passage.
7785 Expressions are extremely light-weight since internally they work like
7786 handles to the actual representation. They really hold nothing more
7787 than a pointer to some other object. What this means in practice is
7788 that whenever you create two @code{ex} and set the second equal to the
7789 first no copying process is involved. Instead, the copying takes place
7790 as soon as you try to change the second. Consider the simple sequence
7795 #include <ginac/ginac.h>
7796 using namespace std;
7797 using namespace GiNaC;
7801 symbol x("x"), y("y"), z("z");
7804 e1 = sin(x + 2*y) + 3*z + 41;
7805 e2 = e1; // e2 points to same object as e1
7806 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
7807 e2 += 1; // e2 is copied into a new object
7808 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
7812 The line @code{e2 = e1;} creates a second expression pointing to the
7813 object held already by @code{e1}. The time involved for this operation
7814 is therefore constant, no matter how large @code{e1} was. Actual
7815 copying, however, must take place in the line @code{e2 += 1;} because
7816 @code{e1} and @code{e2} are not handles for the same object any more.
7817 This concept is called @dfn{copy-on-write semantics}. It increases
7818 performance considerably whenever one object occurs multiple times and
7819 represents a simple garbage collection scheme because when an @code{ex}
7820 runs out of scope its destructor checks whether other expressions handle
7821 the object it points to too and deletes the object from memory if that
7822 turns out not to be the case. A slightly less trivial example of
7823 differentiation using the chain-rule should make clear how powerful this
7828 symbol x("x"), y("y");
7832 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
7833 cout << e1 << endl // prints x+3*y
7834 << e2 << endl // prints (x+3*y)^3
7835 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
7839 Here, @code{e1} will actually be referenced three times while @code{e2}
7840 will be referenced two times. When the power of an expression is built,
7841 that expression needs not be copied. Likewise, since the derivative of
7842 a power of an expression can be easily expressed in terms of that
7843 expression, no copying of @code{e1} is involved when @code{e3} is
7844 constructed. So, when @code{e3} is constructed it will print as
7845 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
7846 holds a reference to @code{e2} and the factor in front is just
7849 As a user of GiNaC, you cannot see this mechanism of copy-on-write
7850 semantics. When you insert an expression into a second expression, the
7851 result behaves exactly as if the contents of the first expression were
7852 inserted. But it may be useful to remember that this is not what
7853 happens. Knowing this will enable you to write much more efficient
7854 code. If you still have an uncertain feeling with copy-on-write
7855 semantics, we recommend you have a look at the
7856 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
7857 Marshall Cline. Chapter 16 covers this issue and presents an
7858 implementation which is pretty close to the one in GiNaC.
7861 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
7862 @c node-name, next, previous, up
7863 @appendixsection Internal representation of products and sums
7865 @cindex representation
7868 @cindex @code{power}
7869 Although it should be completely transparent for the user of
7870 GiNaC a short discussion of this topic helps to understand the sources
7871 and also explain performance to a large degree. Consider the
7872 unexpanded symbolic expression
7874 $2d^3 \left( 4a + 5b - 3 \right)$
7877 @math{2*d^3*(4*a+5*b-3)}
7879 which could naively be represented by a tree of linear containers for
7880 addition and multiplication, one container for exponentiation with base
7881 and exponent and some atomic leaves of symbols and numbers in this
7886 @cindex pair-wise representation
7887 However, doing so results in a rather deeply nested tree which will
7888 quickly become inefficient to manipulate. We can improve on this by
7889 representing the sum as a sequence of terms, each one being a pair of a
7890 purely numeric multiplicative coefficient and its rest. In the same
7891 spirit we can store the multiplication as a sequence of terms, each
7892 having a numeric exponent and a possibly complicated base, the tree
7893 becomes much more flat:
7897 The number @code{3} above the symbol @code{d} shows that @code{mul}
7898 objects are treated similarly where the coefficients are interpreted as
7899 @emph{exponents} now. Addition of sums of terms or multiplication of
7900 products with numerical exponents can be coded to be very efficient with
7901 such a pair-wise representation. Internally, this handling is performed
7902 by most CAS in this way. It typically speeds up manipulations by an
7903 order of magnitude. The overall multiplicative factor @code{2} and the
7904 additive term @code{-3} look somewhat out of place in this
7905 representation, however, since they are still carrying a trivial
7906 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
7907 this is avoided by adding a field that carries an overall numeric
7908 coefficient. This results in the realistic picture of internal
7911 $2d^3 \left( 4a + 5b - 3 \right)$:
7914 @math{2*d^3*(4*a+5*b-3)}:
7920 This also allows for a better handling of numeric radicals, since
7921 @code{sqrt(2)} can now be carried along calculations. Now it should be
7922 clear, why both classes @code{add} and @code{mul} are derived from the
7923 same abstract class: the data representation is the same, only the
7924 semantics differs. In the class hierarchy, methods for polynomial
7925 expansion and the like are reimplemented for @code{add} and @code{mul},
7926 but the data structure is inherited from @code{expairseq}.
7929 @node Package Tools, ginac-config, Internal representation of products and sums, Top
7930 @c node-name, next, previous, up
7931 @appendix Package Tools
7933 If you are creating a software package that uses the GiNaC library,
7934 setting the correct command line options for the compiler and linker
7935 can be difficult. GiNaC includes two tools to make this process easier.
7938 * ginac-config:: A shell script to detect compiler and linker flags.
7939 * AM_PATH_GINAC:: Macro for GNU automake.
7943 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
7944 @c node-name, next, previous, up
7945 @section @command{ginac-config}
7946 @cindex ginac-config
7948 @command{ginac-config} is a shell script that you can use to determine
7949 the compiler and linker command line options required to compile and
7950 link a program with the GiNaC library.
7952 @command{ginac-config} takes the following flags:
7956 Prints out the version of GiNaC installed.
7958 Prints '-I' flags pointing to the installed header files.
7960 Prints out the linker flags necessary to link a program against GiNaC.
7961 @item --prefix[=@var{PREFIX}]
7962 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
7963 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
7964 Otherwise, prints out the configured value of @env{$prefix}.
7965 @item --exec-prefix[=@var{PREFIX}]
7966 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
7967 Otherwise, prints out the configured value of @env{$exec_prefix}.
7970 Typically, @command{ginac-config} will be used within a configure
7971 script, as described below. It, however, can also be used directly from
7972 the command line using backquotes to compile a simple program. For
7976 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
7979 This command line might expand to (for example):
7982 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
7983 -lginac -lcln -lstdc++
7986 Not only is the form using @command{ginac-config} easier to type, it will
7987 work on any system, no matter how GiNaC was configured.
7990 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
7991 @c node-name, next, previous, up
7992 @section @samp{AM_PATH_GINAC}
7993 @cindex AM_PATH_GINAC
7995 For packages configured using GNU automake, GiNaC also provides
7996 a macro to automate the process of checking for GiNaC.
7999 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
8007 Determines the location of GiNaC using @command{ginac-config}, which is
8008 either found in the user's path, or from the environment variable
8009 @env{GINACLIB_CONFIG}.
8012 Tests the installed libraries to make sure that their version
8013 is later than @var{MINIMUM-VERSION}. (A default version will be used
8017 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8018 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8019 variable to the output of @command{ginac-config --libs}, and calls
8020 @samp{AC_SUBST()} for these variables so they can be used in generated
8021 makefiles, and then executes @var{ACTION-IF-FOUND}.
8024 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8025 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8029 This macro is in file @file{ginac.m4} which is installed in
8030 @file{$datadir/aclocal}. Note that if automake was installed with a
8031 different @samp{--prefix} than GiNaC, you will either have to manually
8032 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8033 aclocal the @samp{-I} option when running it.
8036 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8037 * Example package:: Example of a package using AM_PATH_GINAC.
8041 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8042 @c node-name, next, previous, up
8043 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8045 Simply make sure that @command{ginac-config} is in your path, and run
8046 the configure script.
8053 The directory where the GiNaC libraries are installed needs
8054 to be found by your system's dynamic linker.
8056 This is generally done by
8059 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8065 setting the environment variable @env{LD_LIBRARY_PATH},
8068 or, as a last resort,
8071 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8072 running configure, for instance:
8075 LDFLAGS=-R/home/cbauer/lib ./configure
8080 You can also specify a @command{ginac-config} not in your path by
8081 setting the @env{GINACLIB_CONFIG} environment variable to the
8082 name of the executable
8085 If you move the GiNaC package from its installed location,
8086 you will either need to modify @command{ginac-config} script
8087 manually to point to the new location or rebuild GiNaC.
8098 --with-ginac-prefix=@var{PREFIX}
8099 --with-ginac-exec-prefix=@var{PREFIX}
8102 are provided to override the prefix and exec-prefix that were stored
8103 in the @command{ginac-config} shell script by GiNaC's configure. You are
8104 generally better off configuring GiNaC with the right path to begin with.
8108 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8109 @c node-name, next, previous, up
8110 @subsection Example of a package using @samp{AM_PATH_GINAC}
8112 The following shows how to build a simple package using automake
8113 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8117 #include <ginac/ginac.h>
8121 GiNaC::symbol x("x");
8122 GiNaC::ex a = GiNaC::sin(x);
8123 std::cout << "Derivative of " << a
8124 << " is " << a.diff(x) << std::endl;
8129 You should first read the introductory portions of the automake
8130 Manual, if you are not already familiar with it.
8132 Two files are needed, @file{configure.in}, which is used to build the
8136 dnl Process this file with autoconf to produce a configure script.
8138 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8144 AM_PATH_GINAC(0.9.0, [
8145 LIBS="$LIBS $GINACLIB_LIBS"
8146 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8147 ], AC_MSG_ERROR([need to have GiNaC installed]))
8152 The only command in this which is not standard for automake
8153 is the @samp{AM_PATH_GINAC} macro.
8155 That command does the following: If a GiNaC version greater or equal
8156 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8157 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8158 the error message `need to have GiNaC installed'
8160 And the @file{Makefile.am}, which will be used to build the Makefile.
8163 ## Process this file with automake to produce Makefile.in
8164 bin_PROGRAMS = simple
8165 simple_SOURCES = simple.cpp
8168 This @file{Makefile.am}, says that we are building a single executable,
8169 from a single source file @file{simple.cpp}. Since every program
8170 we are building uses GiNaC we simply added the GiNaC options
8171 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8172 want to specify them on a per-program basis: for instance by
8176 simple_LDADD = $(GINACLIB_LIBS)
8177 INCLUDES = $(GINACLIB_CPPFLAGS)
8180 to the @file{Makefile.am}.
8182 To try this example out, create a new directory and add the three
8185 Now execute the following commands:
8188 $ automake --add-missing
8193 You now have a package that can be built in the normal fashion
8202 @node Bibliography, Concept Index, Example package, Top
8203 @c node-name, next, previous, up
8204 @appendix Bibliography
8209 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8212 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8215 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8218 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8221 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8222 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8225 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8226 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8227 Academic Press, London
8230 @cite{Computer Algebra Systems - A Practical Guide},
8231 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8234 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8235 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8238 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8239 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8242 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8247 @node Concept Index, , Bibliography, Top
8248 @c node-name, next, previous, up
8249 @unnumbered Concept Index